____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a_1 x y+b_1 x^2+c_1 x) w_y + (a_2 x y+b_2 x^2+c_2 x) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = D[w[x, y,z], x] +(a1*x*y+b1*x^2+c1*x)*D[w[x, y,z], y] +(a2*x*y+b2*x^2+c2*x)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {-2 \text {a1} \text {b2} x^3-3 \text {a1} \text {c2} x^2+6 \text {a1} z+2 \text {a2} \text {b1} x^3+3 \text {a2} \text {c1} x^2-6 \text {a2} y}{6 \text {a1}},\frac {e^{-\frac {\text {a1} x^2}{2}} (\text {a1} y+\text {b1} x+\text {c1})}{\text {a1}}-\frac {\sqrt {\frac {\pi }{2}} \text {b1} \text {Erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )}{\text {a1}^{3/2}}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := diff(w(x,y,z),x)+(a1*x*y+b1*x^2+c1*x)*diff(w(x,y,z),y)+ (a2*x*y+b2*x^2+c2*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac { \left ( {{\it a1}}^{3/2}y+\sqrt {{\it a1}} \left ( {\it b1}\,x+{\it c1} \right ) \right ) \sqrt {\pi }{{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-1/2\,\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) {\it b1}\,\sqrt {2}\pi }{\sqrt {\pi }{{\it a1}}^{3/2}}},-1/3\,{\frac {1}{{{\it a1}}^{2}} \left ( 3\,{\frac { \left ( {{\it a1}}^{3/2}y+\sqrt {{\it a1}} \left ( {\it b1}\,x+{\it c1} \right ) \right ) {\it a2}\,{{\rm e}^{1/2\,{\it a1}\,{x}^{2}}}{{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}}{\sqrt {{\it a1}}}\sqrt {{\frac {{\it a1}}{\pi }}}}-3/2\,\sqrt {2}\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) {\it a2}\,{\it b1}\, \left ( {\frac {\sqrt {\pi }}{\sqrt {{\it a1}}}\sqrt {{\frac {{\it a1}}{\pi }}}}-1 \right ) {{\rm e}^{1/2\,{\it a1}\,{x}^{2}}}+\sqrt {{\frac {{\it a1}}{\pi }}} \left ( \left ( {\it b2}\,{x}^{3}+3/2\,{x}^{2}{\it c2}-3\,z \right ) {{\it a1}}^{2}-{x}^{2}{\it a2}\, \left ( {\it b1}\,x+3/2\,{\it c1} \right ) {\it a1}-3\,{\it a2}\,{\it b1}\,x \right ) \right ) {\frac {1}{\sqrt {{\frac {{\it a1}}{\pi }}}}}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a_1 x y+b_1 x^2+c_1 x) w_y + (a_2 x z+b_2 x^2+c_2 x) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = D[w[x, y,z], x] +(a1*x*y+b1*x^2+c1*x)*D[w[x, y,z], y] +(a2*x*z+b2*x^2+c2*x)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-\frac {\text {a1} x^2}{2}} (\text {a1} y+\text {b1} x+\text {c1})}{\text {a1}}-\frac {\sqrt {\frac {\pi }{2}} \text {b1} \text {Erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )}{\text {a1}^{3/2}},\frac {e^{-\frac {\text {a2} x^2}{2}} (\text {a2} z+\text {b2} x+\text {c2})}{\text {a2}}-\frac {\sqrt {\frac {\pi }{2}} \text {b2} \text {Erf}\left (\frac {\sqrt {\text {a2}} x}{\sqrt {2}}\right )}{\text {a2}^{3/2}}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := diff(w(x,y,z),x)+(a1*x*y+b1*x^2+c1*x)*diff(w(x,y,z),y)+ (a2*x*z+b2*x^2+c2*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac { \left ( {{\it a1}}^{3/2}y+\sqrt {{\it a1}} \left ( {\it b1}\,x+{\it c1} \right ) \right ) \sqrt {\pi }{{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-1/2\,\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) {\it b1}\,\sqrt {2}\pi }{\sqrt {\pi }{{\it a1}}^{3/2}}},{\frac { \left ( {{\it a2}}^{3/2}z+\sqrt {{\it a2}} \left ( {\it b2}\,x+{\it c2} \right ) \right ) \sqrt {\pi }{{\rm e}^{-1/2\,{x}^{2}{\it a2}}}-1/2\,{\it b2}\,\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a2}}x \right ) \sqrt {2}\pi }{\sqrt {\pi }{{\it a2}}^{3/2}}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a_1 x y+b_1 x^2+c_1 x) w_y + (a_2 y z+b_2 y^2+c_2 y) w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = D[w[x, y,z], x] +(a1*x*y+b1*x^2+c1*x)*D[w[x, y,z], y] +(a2*y*z+b2*y^2+c2*y)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {\$Aborted} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := diff(w(x,y,z),x)+(a1*x*y+b1*x^2+c1*x)*diff(w(x,y,z),y)+ (a2*y*z+b2*y^2+c2*y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac { \left ( {{\it a1}}^{3/2}y+\sqrt {{\it a1}} \left ( {\it b1}\,x+{\it c1} \right ) \right ) \sqrt {\pi }{{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-1/2\,\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) {\it b1}\,\sqrt {2}\pi }{\sqrt {\pi }{{\it a1}}^{3/2}}},-1/2\,\int ^{x}\!2\,{\frac { \left ( -2\,{\it \_f}\,{\it b1}\,{\it b2}+{\it c2}\,{\it a1}-2\,{\it b2}\,{\it c1} \right ) \left ( \pi \, \left ( {{\it a1}}^{3/2}y+\sqrt {{\it a1}} \left ( {\it b1}\,x+{\it c1} \right ) \right ) {{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-1/2\,\sqrt {2}{\pi }^{3/2}{\it b1}\, \left ( \erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) -\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}{\it \_f} \right ) \right ) \right ) {\it a1}\,{{\rm e}^{1/2\,{\it a1}\,{{\it \_f}}^{2}}}+\pi \,\sqrt {{\it a1}}{{\rm e}^{{\it a1}\,{{\it \_f}}^{2}}} \left ( {{\it a1}}^{3/2}y+\sqrt {{\it a1}} \left ( {\it b1}\,x+{\it c1} \right ) \right ) ^{2}{\it b2}\, \left ( {{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}} \right ) ^{2}-{\pi }^{3/2} \left ( \erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) -\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}{\it \_f} \right ) \right ) \sqrt {{\it a1}}\sqrt {2}{{\rm e}^{{\it a1}\,{{\it \_f}}^{2}}}{\it b1}\, \left ( {{\it a1}}^{3/2}y+\sqrt {{\it a1}} \left ( {\it b1}\,x+{\it c1} \right ) \right ) {\it b2}\,{{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-\pi \, \left ( -1/2\, \left ( \erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}{\it \_f} \right ) \right ) ^{2}{{\it b1}}^{2}{\it b2}\,{{\rm e}^{{\it a1}\,{{\it \_f}}^{2}}}\pi \,\sqrt {{\it a1}}+\sqrt {{\it a1}}{{\rm e}^{{\it a1}\,{{\it \_f}}^{2}}}\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) \erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}{\it \_f} \right ) \pi \,{{\it b1}}^{2}{\it b2}-1/2\,\sqrt {{\it a1}}{{\rm e}^{{\it a1}\,{{\it \_f}}^{2}}} \left ( \erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) \right ) ^{2}\pi \,{{\it b1}}^{2}{\it b2}+ \left ( {\it \_f}\,{\it b1}+{\it c1} \right ) \left ( -{\it b2}\, \left ( {\it \_f}\,{\it b1}+{\it c1} \right ) {{\it a1}}^{3/2}+{{\it a1}}^{5/2}{\it c2} \right ) \right ) }{{{\it a1}}^{7/2}\pi }{{\rm e}^{-1/2\,{\frac {{\it a2}\,\int \! \left ( \left ( 2\,{{\it a1}}^{3/2}y+2\,\sqrt {{\it a1}} \left ( {\it b1}\,x+{\it c1} \right ) \right ) {{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-\sqrt {\pi }{\it b1}\,\sqrt {2} \left ( \erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) -\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}{\it \_f} \right ) \right ) \right ) {{\rm e}^{1/2\,{\it a1}\,{{\it \_f}}^{2}}}-2\,\sqrt {{\it a1}} \left ( {\it \_f}\,{\it b1}+{\it c1} \right ) \,{\rm d}{\it \_f}}{{{\it a1}}^{3/2}}}}}}{d{\it \_f}}+z{{\rm e}^{-1/2\,{\frac {1}{\sqrt {\pi }}\int ^{x}\!2\,{\frac {{\it a2}\, \left ( {{\rm e}^{1/2\,{\it a1}\,{{\it \_b}}^{2}}} \left ( {{\it a1}}^{3/2}y+\sqrt {{\it a1}} \left ( {\it b1}\,x+{\it c1} \right ) \right ) \sqrt {\pi }{{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-1/2\,{\it b1}\,\sqrt {2}\pi \, \left ( \erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) -\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}{\it \_b} \right ) \right ) {{\rm e}^{1/2\,{\it a1}\,{{\it \_b}}^{2}}}-\sqrt {\pi }\sqrt {{\it a1}} \left ( {\it \_b}\,{\it b1}+{\it c1} \right ) \right ) }{{{\it a1}}^{3/2}}}{d{\it \_b}}}}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a_1 x y+b_1 y^2) w_y + (a_2 x z+b_2 z^2) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = D[w[x, y,z], x] +(a1*x+b1*y^2)*D[w[x, y,z], y] +(a2*x*z+b2*z^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (-\frac {2 \left (\text {b1} x y \text {BesselJ}\left (\frac {1}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )+\sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2} \text {BesselJ}\left (-\frac {2}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )\right )}{(2 \text {b1} x y+1) \text {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )+\sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2} \text {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )-\sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2} \text {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )},\frac {\sqrt {\frac {\pi }{2}} \text {b2} \text {Erfi}\left (\frac {\sqrt {\text {a2}} x}{\sqrt {2}}\right )}{\sqrt {\text {a2}}}+\frac {e^{\frac {\text {a2} x^2}{2}}}{z}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := diff(w(x,y,z),x)+(a1*x+b1*y^2)*diff(w(x,y,z),y)+ (a2*x*z+b2*z^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y\AiryBi \left ( -\sqrt [3]{{\it a1}\,{\it b1}}x \right ) {\it b1}-\sqrt [3]{{\it a1}\,{\it b1}}\AiryBi \left ( 1,-\sqrt [3]{{\it a1}\,{\it b1}}x \right ) }{-y\AiryAi \left ( -\sqrt [3]{{\it a1}\,{\it b1}}x \right ) {\it b1}+\sqrt [3]{{\it a1}\,{\it b1}}\AiryAi \left ( 1,-\sqrt [3]{{\it a1}\,{\it b1}}x \right ) }},{\frac {\sqrt {\pi }\erf \left ( 1/2\,\sqrt {-2\,{\it a2}}x \right ) z{\it b2}+\sqrt {-2\,{\it a2}}{{\rm e}^{1/2\,{x}^{2}{\it a2}}}}{\sqrt {-2\,{\it a2}}z}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + x z w_y - x y w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*D[w[x, y,z], x] +x*z*D[w[x, y,z], y] -x*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (y \sin \left (\frac {x^2}{2 a}\right )+z \cos \left (\frac {x^2}{2 a}\right ),y \cos \left (\frac {x^2}{2 a}\right )-z \sin \left (\frac {x^2}{2 a}\right )\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*diff(w(x,y,z),x)+x*z*diff(w(x,y,z),y)- x*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {y}^{2}+{z}^{2},-2\,a\arctan \left ( {\frac {y}{z}} \right ) +{x}^{2} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ c x w_x + c y w_y +(a x^2+b y^2) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = c*x*D[w[x, y,z], x] +c*y*D[w[x, y,z], y] +(a*x^2+b*y^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},-\frac {a x^2+b y^2-2 c z}{2 c}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := c*x*diff(w(x,y,z),x)+c*y*diff(w(x,y,z),y)+(a*x^2+b*y^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}},1/2\,{\frac {-a{x}^{2}-b{y}^{2}+2\,cz}{c}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ c z w_x -a(2 a x-b)y w_y +a (2 a x-b)z w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = c*z*D[w[x, y,z], x] -a*(2*a*x-b)*y*D[w[x, y,z], y] +a*(2*a*x-b)*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (c y z,\frac {-a^2 x^2+a b x+c z}{c}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := c*z*diff(w(x,y,z),x)-a*(2*a*x-b)*diff(w(x,y,z),y)+a*(2*a*x-b)*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-{a}^{2}{x}^{2}+axb+cz}{c}},\ln \left ( cz \right ) +y \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a c x^2 w_x -a c x y w_y -b^2 y^2 w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*c*x^2*D[w[x, y,z], x] -a*c*x*y*D[w[x, y,z], y] -b^2*y^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (x y,z-\frac {b^2 y^2}{3 a c x}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*c*x^2*diff(w(x,y,z),x) -a*c*x*y*diff(w(x,y,z),y)-b^2*y^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( yx,1/3\,{\frac {3\,zac{x}^{3}-{b}^{2}{x}^{2}{y}^{2}}{ac{x}^{3}}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^2 w_x +b y^2 w_y +c z^2 w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*x^2*D[w[x, y,z], x] +b*y^2*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*x^2*diff(w(x,y,z),x) +b*y^2*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ax-by}{yax}},{\frac {ax-cz}{zax}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a b x^2 w_x +c z^2 w_y +2 a b x z w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*b*x^2*D[w[x, y,z], x] +c*z^2*D[w[x, y,z], y] +2*a*b*x*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {z}{x^2},y-\frac {c z^2}{3 a b x}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*b*x^2*diff(w(x,y,z),x) +c*z^2*diff(w(x,y,z),y)+2*a*b*x*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {z}{{x}^{2}}},1/3\,{\frac {3\,abxy-c{z}^{2}}{axb}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b c x y w_x +a^2 c x^2 w_y - b y (2 a x+c z)w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*c*x*y*D[w[x, y,z], x] +a^2*c*x^2*D[w[x, y,z], y] -b*y*(2*a*x+c*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {b y^2-a^2 x^2}{2 b},\frac {x (a x+c z)}{c}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*c*x*y*diff(w(x,y,z),x) +a^2*c*x^2*diff(w(x,y,z),y)-b*y*(2*a*x+c*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-{a}^{2}{x}^{2}+b{y}^{2}}{b}},{\frac { \left ( ax+cz \right ) x}{c}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b c x y w_x +c^2 y z w_y + b^2 y^2 w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*c*x*y*D[w[x, y,z], x] +c^2*y*z*D[w[x, y,z], y] +b^2*y^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {i \left (b \left (x^2-1\right ) y-c \left (x^2+1\right ) z\right )}{2 b x},\frac {b \left (x^2+1\right ) y-c \left (x^2-1\right ) z}{2 b x}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*c*x*y*diff(w(x,y,z),x) +c^2*y*z*diff(w(x,y,z),y)+b^2*y^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-{y}^{2}{b}^{2}+{c}^{2}{z}^{2}}{{c}^{2}}},x \left ( {\it csgn} \left ( b \right ) by+cz \right ) ^{-{\it csgn} \left ( b \right ) } \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x y w_x +y(y-a)w_y +z(y-a) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = x*y*D[w[x, y,z], x] +y*(y-a)*D[w[x, y,z], y] +z*(y-a)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {y-a}{x},\frac {z}{y}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := x*y*diff(w(x,y,z),x) +y*(y-a)*diff(w(x,y,z),y)+z*(y-a)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y-a}{x}},{\frac {z}{y}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b y^2 w_x -a x y w_y +c x z w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*y^2*D[w[x, y,z], x] -a*x*y*D[w[x, y,z], y] +c*x*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {a x^2+b y^2}{2 b},z \left (-b y^2\right )^{\frac {c}{2 a}}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*y^2*diff(w(x,y,z),x) -a*x*y*diff(w(x,y,z),y)+c*x*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {a{x}^{2}+b{y}^{2}}{b}},z \left ( -b{y}^{2} \right ) ^{1/2\,{\frac {c}{a}}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ c x z w_x + 2 a x y w_y -(2 a x+c z) z w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = c*x*z*D[w[x, y,z], x] +2*a*x*y*D[w[x, y,z], y] -(2*a*x+c*z)*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (-c x y z,x \left (\frac {a x}{c}+z\right )\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := c*x*z*diff(w(x,y,z),x) +2*a*x*y*diff(w(x,y,z),y)-(2*a*x+c*z)*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac { \left ( ax+cz \right ) x}{c}},-cxyz \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.16, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ c x z w_x + c y z w_y +a b x y w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = c*x*z*D[w[x, y,z], x] +c*y*z*D[w[x, y,z], y] +a*b*x*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {c z^2-a b x y}{2 c}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := c*x*z*diff(w(x,y,z),x) +c*y*z*diff(w(x,y,z),y)+a*b*x*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}},{\frac {-abxy+c{z}^{2}}{c}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.17, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ c x z w_x - c y z w_y +(b y^2-a x) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = c*x*z*D[w[x, y,z], x] -c*y*z*D[w[x, y,z], y] +(b*y^2-a*x)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (x y,\frac {2 a x+b y^2+c z^2}{2 c}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := c*x*z*diff(w(x,y,z),x)-c*y*z*diff(w(x,y,z),y)+(b*y^2-a*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( yx,{\frac {b{x}^{2}{y}^{2}+{z}^{2}c{x}^{2}+2\,a{x}^{3}}{c{x}^{2}}} \right ) \]
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Added April 14, 2019.
Problem Chapter 6.2.2.18, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ c x z w_x - c y z w_y +(a x^2+b y^2 ) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = c*x*z*D[w[x, y,z], x] -c*y*z*D[w[x, y,z], y] +(a*x^2+b*y^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (x y,\frac {-a x^2+b y^2+c z^2}{2 c}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := c*x*z*diff(w(x,y,z),x)-c*y*z*diff(w(x,y,z),y)+(a*x^2+b*y^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( yx,{\frac {-a{x}^{2}+b{y}^{2}+c{z}^{2}}{c}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.19, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x z w_x + y z w_y +(a x^2+a y^2+ b z^2 ) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = x*z*D[w[x, y,z], x] +y*z*D[w[x, y,z], y] +(a*x^2+a*y^2+b*z^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {x^{-2 b} \left (a \left (x^2+y^2\right )+(b-1) z^2\right )}{b-1}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := x*z*diff(w(x,y,z),x)+y*z*diff(w(x,y,z),y)+(a*x^2+a*y^2+b*z^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}},{\frac {{x}^{-2\,b} \left ( \left ( {x}^{2}+{y}^{2} \right ) a+{z}^{2} \left ( b-1 \right ) \right ) }{b-1}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.20, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ 2 c x z w_x + 2 c y z w_y +(c z^2-a x^2- b y^2 ) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = 2*c*x*z*D[w[x, y,z], x] +2*c*y*z*D[w[x, y,z], y] +(c*z^2-a*x^2-b*y^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {a x^2+b y^2+c z^2}{c x}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := 2*c*x*z*diff(w(x,y,z),x)+2*c*y*z*diff(w(x,y,z),y)+(c*z^2-a*x^2-b*y^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}},{\frac {a{x}^{2}+b{y}^{2}+c{z}^{2}}{cx}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.21, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b c y z w_x + a c x z w_y + a b x y w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*c*y*z*D[w[x, y,z], x] +a*c*x*z*D[w[x, y,z], y] +a*b*x*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {b y^2-a x^2}{2 b},\frac {c z^2-a x^2}{2 c}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*c*y*z*diff(w(x,y,z),x)+a*c*x*z*diff(w(x,y,z),y)+a*b*x*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-a{x}^{2}+b{y}^{2}}{b}},{\frac {-a{x}^{2}+c{z}^{2}}{c}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.22, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b c (x^2-a^2) w_x + c(b x y+a c z ) w_y + b(c x z + a b y) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*c*(x^2-a^2)*D[w[x, y,z], x] +c*(b*x*y+a*c*z)*D[w[x, y,z], y] +b*(c*x*z + a*b*y)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (-\frac {a c z+b x y}{a^2 b-b x^2},\frac {a (a b y+c x z)}{b \left (a^2-x^2\right )}\right )\right \}\right \} \]
Maple ✗
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*c*(x^2-a^2)*diff(w(x,y,z),x)+c*(b*x*y+a*c*z)*diff(w(x,y,z),y)+b*(c*x*z + a*b*y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ \text { sol=() } \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.23, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b x (b y +c) w_x + (b^2 y^2-a c x ) w_y + b^2 y z w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*x*(b*y+c)*D[w[x, y,z], x] + (b^2*y^2-a*c*x )*D[w[x, y,z], y] + b^2*y*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*x*(b*y+c)*diff(w(x,y,z),x)+(b^2*y^2-a*c*x )*diff(w(x,y,z),y)+b^2*y*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{3\,ax+3\,by} \left ( \left ( -ax-by \right ) \ln \left ( -9\,{\frac { \left ( ax+by \right ) \left ( ax-c \right ) }{x \left ( by+c \right ) }} \right ) + \left ( ax+by \right ) \ln \left ( {\frac {-9\,ax+9\,c}{2\,by+2\,c}} \right ) +by+c \right ) },z{{\rm e}^{-1/9\,\int ^{x}\!{\frac {1}{{\it \_a}\, \left ( {\it \_a}\,a-c \right ) } \left ( 2\,{{\rm e}^{\RootOf \left ( 2\,\ln \left ( -9\,{\frac { \left ( ax+by \right ) \left ( ax-c \right ) }{x \left ( by+c \right ) }} \right ) {{\rm e}^{{\it \_Z}}}ax+2\,\ln \left ( -9\,{\frac { \left ( ax+by \right ) \left ( ax-c \right ) }{x \left ( by+c \right ) }} \right ) {{\rm e}^{{\it \_Z}}}by-2\,{{\rm e}^{{\it \_Z}}}\ln \left ( {\frac { \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) \left ( {\it \_a}\,a-c \right ) }{{\it \_a}}} \right ) ax-2\,{{\rm e}^{{\it \_Z}}}\ln \left ( {\frac { \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) \left ( {\it \_a}\,a-c \right ) }{{\it \_a}}} \right ) by-2\,{{\rm e}^{{\it \_Z}}}\ln \left ( -9/2\,{\frac {ax-c}{by+c}} \right ) ax-2\,{{\rm e}^{{\it \_Z}}}\ln \left ( -9/2\,{\frac {ax-c}{by+c}} \right ) by+2\,{\it \_Z}\,{{\rm e}^{{\it \_Z}}}ax+2\,{{\rm e}^{{\it \_Z}}}{\it \_Z}\,by-9\,\ln \left ( -9\,{\frac { \left ( ax+by \right ) \left ( ax-c \right ) }{x \left ( by+c \right ) }} \right ) ax-9\,\ln \left ( -9\,{\frac { \left ( ax+by \right ) \left ( ax-c \right ) }{x \left ( by+c \right ) }} \right ) yb-2\,{{\rm e}^{{\it \_Z}}}by+9\,\ln \left ( {\frac { \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) \left ( {\it \_a}\,a-c \right ) }{{\it \_a}}} \right ) ax+9\,\ln \left ( {\frac { \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) \left ( {\it \_a}\,a-c \right ) }{{\it \_a}}} \right ) by+9\,\ln \left ( -9/2\,{\frac {ax-c}{by+c}} \right ) ax+9\,\ln \left ( -9/2\,{\frac {ax-c}{by+c}} \right ) by-9\,ax{\it \_Z}-9\,by{\it \_Z}-2\,c{{\rm e}^{{\it \_Z}}}-9\,ax+9\,c \right ) }}c+9\,{\it \_a}\,a-9\,c \right ) }{d{\it \_a}}}} \right ) \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.24, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x (b y -c z) w_x + y(c z-a x) w_y + z(a x - b y) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = x*(b*y -c*z)*D[w[x, y,z], x] + y*(c*z-a*x)*D[w[x, y,z], y] + z*(a*x - b*y)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x y z}{b},\frac {a x+b y+c z}{c}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := x*(b*y -c*z)*diff(w(x,y,z),x)+ y*(c*z-a*x)*diff(w(x,y,z),y)+z*(a*x - b*y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\frac {{x}^{{\it \_C2}}{{\rm e}^{{\it \_C2}}}{\it \_C5}\,{y}^{{\it \_C2}}{\it \_C4}\,{z}^{{\it \_C2}}}{{{\rm e}^{{\it \_C1}\,x}}{\it \_C3}} \left ( {{\rm e}^{{\frac {by{\it \_C1}}{a}}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {cz{\it \_C1}}{a}}}} \right ) ^{-1}} \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.25, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a(y+\beta )(z+\gamma ) w_x -b(x+\alpha )(z+\gamma ) w_y - c(x+\alpha )(y+\beta ) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*(y+beta)*(z+gamma)*D[w[x, y,z], x] -b*(x+alpha)*(z+gamma)*D[w[x, y,z], y] - c*(x+alpha)*(y+beta)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {a y (2 \beta +y)+2 \alpha b x+b x^2}{2 a},\frac {a z (2 \gamma +z)+2 \alpha c x+c x^2}{2 a}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*(y+beta)*(z+gamma)*diff(w(x,y,z),x)-b*(x+alpha)*(z+gamma)*diff(w(x,y,z),y)- c*(x+alpha)*(y+beta)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={{{\rm e}^{{\it \_C3}\,\alpha \,x}}{{\rm e}^{1/2\,{x}^{2}{\it \_C3}}}{{\rm e}^{{\it \_C1}\,\beta \,y}}{{\rm e}^{1/2\,{\it \_C1}\,{y}^{2}}}{\it \_C4}\,{\it \_C2}\,{\it \_C5}{{\rm e}^{{\frac {a\gamma \,{\it \_C3}\,z}{c}}}}{{\rm e}^{-1/2\,{\frac {{\it \_C1}\,b{z}^{2}}{c}}}}{{\rm e}^{1/2\,{\frac {a{\it \_C3}\,{z}^{2}}{c}}}} \left ( {{\rm e}^{{\frac {{\it \_C1}\,b\gamma \,z}{c}}}} \right ) ^{-1}} \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.26, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b c (a c x z + b^2 y^2) w_x +a c (b c y z-2 a^2 x^2)w_y - a b (2 a b x y+c^2 z^2) w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*c*(a*c*x*z + b^2*y^2)*D[w[x, y,z], x] +a*c*(b*c*y*z-2*a^2*x^2)*D[w[x, y,z], y] - a*b*(2*a*b*x*y+c^2*z^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✗
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*c*(a*c*x*z + b^2*y^2)*diff(w(x,y,z),x)+a*c*(b*c*y*z-2*a^2*x^2)*diff(w(x,y,z),y)- a*b*(2*a*b*x*y+c^2*z^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ \text { sol=() } \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.27, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a(y^2+z^2) w_x +x(b z-a y)w_y -x(b y + a z) w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*(y^2+z^2)*D[w[x, y,z], x] +x*(b*z-a*y)*D[w[x, y,z], y] -x*(b*y + a*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*(y^2+z^2)*diff(w(x,y,z),x)+x*(b*z-a*y)*diff(w(x,y,z),y)-x*(b*y + a*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={{\it \_C1}\,{{\rm e}^{1/2\,{\it \_c}_{{1}}{x}^{2}}}{\it \_F5} \left ( 1/2\,{\frac {1}{b} \left ( 2\,a\arctan \left ( {\frac {z}{y}} \right ) -b\ln \left ( {y}^{2}+{z}^{2} \right ) \right ) } \right ) \left ( {{\rm e}^{a{\it \_c}_{{1}}\int ^{y}\!{{\it \_a} \left ( \cos \left ( \RootOf \left ( -2\,a\arctan \left ( {\frac {z}{y}} \right ) -b\ln \left ( {\frac {{{\it \_a}}^{2}}{\cos \left ( 2\,{\it \_Z} \right ) +1}} \right ) -\ln \left ( 2 \right ) b+b\ln \left ( {y}^{2}+{z}^{2} \right ) +2\,{\it \_Z}\,a \right ) \right ) \right ) ^{-1} \left ( \sin \left ( \RootOf \left ( -2\,a\arctan \left ( {\frac {z}{y}} \right ) -b\ln \left ( {\frac {{{\it \_a}}^{2}}{\cos \left ( 2\,{\it \_Z} \right ) +1}} \right ) -\ln \left ( 2 \right ) b+b\ln \left ( {y}^{2}+{z}^{2} \right ) +2\,{\it \_Z}\,a \right ) \right ) b-a\cos \left ( \RootOf \left ( -2\,a\arctan \left ( {\frac {z}{y}} \right ) -b\ln \left ( {\frac {{{\it \_a}}^{2}}{\cos \left ( 2\,{\it \_Z} \right ) +1}} \right ) -\ln \left ( 2 \right ) b+b\ln \left ( {y}^{2}+{z}^{2} \right ) +2\,{\it \_Z}\,a \right ) \right ) \right ) ^{-1}}{d{\it \_a}}}} \right ) ^{-1}} \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.28, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b(b y + c z)^2 w_x - a x(b y + 2 c z)w_y +a b x z w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*(b*y + c*z)^2*D[w[x, y,z], x] - a*x*(b*y + 2*c*z)*D[w[x, y,z], y] +a*b*x*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {2 \left (a x^2+c^2 z^2\right )}{b},\log (z (b y+c z))\right )\right \},\left \{w(x,y,z)\to c_1\left (\frac {2 \left (a x^2+c^2 z^2\right )}{b},\log (z (c z-b y))\right )\right \},\left \{w(x,y,z)\to c_1\left (\frac {2 \left (a x^2+(b y-c z)^2\right )}{b},\log (z (c z-b y))\right )\right \},\left \{w(x,y,z)\to c_1\left (\frac {2 \left (a x^2+(b y+c z)^2\right )}{b},\log (z (b y+c z))\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*(b*y + c*z)^2*diff(w(x,y,z),x)- a*x*(b*y + 2*c*z)*diff(w(x,y,z),y)+a*b*x*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_C1}\,{{\rm e}^{1/2\,{\it \_c}_{{1}}{x}^{2}}}{\it \_F5} \left ( {\frac {z \left ( by+cz \right ) }{b}} \right ) {{\rm e}^{1/2\,{\frac {{b}^{2}{\it \_c}_{{1}}{y}^{2}}{a}}}}{{\rm e}^{1/2\,{\frac {b{\it \_c}_{{1}}ycz}{a}}}} \]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.29, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (f_0 x - f_1) w_x + (f_0 y-f_2) w_y + (f_0 z -f_3) w_z= 0 \] Where \[ f_n = a_n + b_n x + c_n y+ d_n z \]
Mathematica ✗
ClearAll[w, x, y, z, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; f[n_]:= a[n] + b[n]*x + c[n]*y+ d[n]*z; pde =(f[0]*x - f[1])*D[w[x, y,z], x] +(f[0]*y-f[2])*D[w[x, y,z], y] +(f[0]*z -f[3])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✗
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); f:= n -> a[n] + b[n]*x + c[n]*y+ d[n]*z; pde := (f(0)*x - f(1))*diff(w(x,y,z),x)+(f(0)*y-f(2))*diff(w(x,y,z),y)+(f(0)*z -f(3))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ \text { sol=() } \]