[next] [prev] [prev-tail] [tail] [up]

150 HFOPDE, chapter 6.2.1

150.1 Problem 1
150.2 Problem 2
150.3 Problem 3
150.4 Problem 4
150.5 Problem 5
150.6 Problem 6
150.7 Problem 7
150.8 Problem 8
150.9 Problem 9
150.10 Problem 10
150.11 Problem 11
150.12 Problem 12
150.13 Problem 13
150.14 Problem 14
150.15 Problem 15
150.16 Problem 16
150.17 Problem 17
150.18 Problem 18
150.19 Problem 19
150.20 Problem 20
150.21 Problem 21

____________________________________________________________________________________

150.1 Problem 1

problem number 1176

Added April 13, 2019.

Problem Chapter 6.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y + c w_z= 0 \]

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]; 
 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] + b*D[w[x, y,z], y] +c*D[w[x,y,z],z]==0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; 
 sol = Simplify[sol];

\[ \left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )\right \}\right \} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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)+ b*diff(w(x,y,z),y) + c*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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {za-cx}{a}} \right ) \]

____________________________________________________________________________________

150.2 Problem 2

problem number 1177

Added April 13, 2019.

Problem Chapter 6.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a x w_y + b y w_z= 0 \]

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]; 
 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] + a*x*D[w[x, y,z], y] +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];

\[ \left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a x^2}{2},\frac {1}{3} a b x^3-b x y+z\right )\right \}\right \} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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)+ a*x*diff(w(x,y,z),y) + 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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( -1/2\,a{x}^{2}+y,1/3\,x \left ( a{x}^{2}-3\,y \right ) b+z \right ) \]

____________________________________________________________________________________

150.3 Problem 3

problem number 1178

Added April 13, 2019.

Problem Chapter 6.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b y w_y + c z w_z= 0 \]

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]; 
 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] + b*y*D[w[x, y,z], y] +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];

\[ \left \{\left \{w(x,y,z)\to c_1\left (y e^{-\frac {b x}{a}},z e^{-\frac {c x}{a}}\right )\right \}\right \} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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)+ b*y*diff(w(x,y,z),y) + 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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( y{{\rm e}^{-{\frac {bx}{a}}}},z{{\rm e}^{-{\frac {cx}{a}}}} \right ) \]

____________________________________________________________________________________

150.4 Problem 4

problem number 1179

Added April 13, 2019.

Problem Chapter 6.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a z w_y + b y w_z= 0 \]

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]; 
 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] + a*z*D[w[x, y,z], y] +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];

\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )\right \}\right \} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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)+ a*z*diff(w(x,y,z),y) + 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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{z}^{2}a-b{y}^{2}}{a}},-{\frac {1}{\sqrt {ab}} \left ( -x\sqrt {ab}+\ln \left ( {\frac {aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ab}}{\sqrt {ab}}} \right ) \right ) } \right ) \]

____________________________________________________________________________________

150.5 Problem 5

problem number 1180

Added April 13, 2019.

Problem Chapter 6.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + a y w_y + b z w_z= 0 \]

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]; 
 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*D[w[x, y,z], x] + a*y*D[w[x, y,z], y] +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];

\[ \left \{\left \{w(x,y,z)\to c_1\left (y x^{-a},z x^{-b}\right )\right \}\right \} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*diff(w(x,y,z),x)+ a*y*diff(w(x,y,z),y) + 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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( y{x}^{-a},z{x}^{-b} \right ) \]

____________________________________________________________________________________

150.6 Problem 6

problem number 1181

Added April 13, 2019.

Problem Chapter 6.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + a z w_y + b y w_z= 0 \]

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]; 
 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*D[w[x, y,z], x] + a*z*D[w[x, y,z], y] +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];

\[ \left \{\left \{w(x,y,z)\to c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right \}\right \} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*diff(w(x,y,z),x)+ a*z*diff(w(x,y,z),y) + 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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{z}^{2}a-b{y}^{2}}{a}},x \left ( \sqrt {ab}y+za \right ) ^{-{\frac {\sqrt {ab}}{ab}}} \right ) \]

____________________________________________________________________________________

150.7 Problem 7

problem number 1182

Added April 13, 2019.

Problem Chapter 6.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + (a x+b y) w_y + (\alpha x+\beta y+\gamma z) w_z= 0 \]

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]; 
 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*D[w[x, y,z], x] + (a*x+b*y)*D[w[x, y,z], y] +(alpha*x+beta*y+gamma*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];

\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {x^{-b} (a x+(b-1) y)}{b-1},\frac {x^{-\gamma } (-a \beta x+\alpha x (b-\gamma )-(\gamma -1) (-b z+\beta y+\gamma z))}{(\gamma -1) (b-\gamma )}\right )\right \}\right \} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*diff(w(x,y,z),x)+ (a*x+b*y)*diff(w(x,y,z),y) + (alpha*x+beta*y+gamma*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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac { \left ( y \left ( b-1 \right ) +ax \right ) {x}^{-b}}{b-1}},{\frac {- \left ( -\gamma +b \right ) \left ( a\beta -\alpha \,b+\alpha \right ) {x}^{1-\gamma }- \left ( z \left ( b-1 \right ) \gamma -{b}^{2}z+ \left ( \beta \,y+z \right ) b+\beta \, \left ( ax-y \right ) \right ) \left ( -1+\gamma \right ) {x}^{-\gamma }}{ \left ( -1+\gamma \right ) \left ( b-1 \right ) \left ( -\gamma +b \right ) }} \right ) \]

____________________________________________________________________________________

150.8 Problem 8

problem number 1183

Added April 13, 2019.

Problem Chapter 6.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a b x w_x + (a y+b z) ( b w_y -a w_z)= 0 \]

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]; 
 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*D[w[x, y,z], x] + (a*y+b*z)*(b*D[w[x, 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];

\[ \text {Failed} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*diff(w(x,y,z),x)+ (a*y+b*z)*(b*diff(w(x,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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya+bz}{b}},x{{\rm e}^{-{\frac {ya}{ya+bz}}}} \right ) \]

____________________________________________________________________________________

150.9 Problem 9

problem number 1184

Added April 13, 2019.

Problem Chapter 6.2.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a b x w_x + b (a y+b z) w_y + a(a y-b z) w_z= 0 \]

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]; 
 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*D[w[x, y,z], x] + b*(a*y+b*z)*D[w[x, y,z], y] +a*(a*y-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];

\[ \text {Failed} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*diff(w(x,y,z),x)+ b*(a*y+b*z)*diff(w(x,y,z),y) + a*(a*y-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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( -{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}},x \left ( {1 \left ( {\frac {\sqrt {2}{a}^{2}y}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}+ \left ( {\frac {ya}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}+{\frac {bz}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}} \right ) \sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}} \right ) ^{-1/2\,{\frac {a\sqrt {2}}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}{\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}}} \right ) \]

____________________________________________________________________________________

150.10 Problem 10

problem number 1185

Added April 13, 2019.

Problem Chapter 6.2.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ b^2 c y w_x + a^2 c x w_y - a b(a x+b y) w_z= 0 \]

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]; 
 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^2*c*y*D[w[x, y,z], x] + a^2*c*x*D[w[x, y,z], y] -a*b*(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];

\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {1}{2} \left (y^2-\frac {a^2 x^2}{b^2}\right ),\frac {a x+b y+c z}{c}\right )\right \}\right \} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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^2*c*y*diff(w(x,y,z),x)+ a^2*c*x*diff(w(x,y,z),y) - a*b*(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))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-{a}^{2}{x}^{2}+{y}^{2}{b}^{2}}{{b}^{2}}},{\frac {ax+by+cz}{c}} \right ) \]

____________________________________________________________________________________

150.11 Problem 11

problem number 1186

Added April 13, 2019.

Problem Chapter 6.2.1.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ c z w_x + (a x +b y) w_y +(a x+b y+c z) w_z= 0 \]

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]; 
 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*x+b*y)*D[w[x, y,z], y] +(a*x+b*y+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];

\[ \text {Failed} \]

Maple ✗ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*x+b*y)*diff(w(x,y,z),y) + (a*x+b*y+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;

\[ \text { sol=() } \]

____________________________________________________________________________________

150.12 Problem 12

problem number 1187

Added April 13, 2019.

Problem Chapter 6.2.1.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ b^2 c z w_x - a^2 c x w_y + a b^2 y w_z= 0 \]

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]; 
 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^2*c*z*D[w[x, y,z], x] - a^2*c*x*D[w[x, y,z], y] +a*b^2*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];

\[ \text {Failed} \]

Maple ✗ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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^2*c*z*diff(w(x,y,z),x)-a^2*c*x*diff(w(x,y,z),y) + a*b^2*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;

\[ \text { sol=() } \]

____________________________________________________________________________________

150.13 Problem 13

problem number 1188

Added April 13, 2019.

Problem Chapter 6.2.1.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (x+a) w_x + (y+b) x w_y + (z+c) w_z= 0 \]

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]; 
 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+a)*D[w[x, y,z], x] + (y+b)*D[w[x, y,z], y] +(z+c)*D[w[x,y,z],z]==0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; 
 sol = Simplify[sol];

\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {b+y}{a+x},\frac {c+z}{a+x}\right )\right \}\right \} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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+a)*diff(w(x,y,z),x)+(y+b)*diff(w(x,y,z),y) + (z+c)*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;

\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {b+y}{x+a}},{\frac {z+c}{x+a}} \right ) \]

____________________________________________________________________________________

150.14 Problem 14

problem number 1189

Added April 13, 2019.

Problem Chapter 6.2.1.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ 2 b c(a x-b y) w_x -a c(a x-b y-c z)w_y - a b (a x -b y-3 c z) w_z= 0 \]

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]; 
 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*b*c*(a*x-b*y)*D[w[x, y,z], x] -a*c*(a*x-b*y-c*z)*D[w[x, y,z], y] - a*b*(a*x -b*y-3*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];

\[ \text {\$Aborted} \]

Maple ✗ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*b*c*(a*x-b*y)*diff(w(x,y,z),x)-a*c*(a*x-b*y-c*z)*diff(w(x,y,z),y)- a*b*(a*x -b*y-3*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;

\[ \text { sol=() } \]

____________________________________________________________________________________

150.15 Problem 15

problem number 1190

Added April 13, 2019.

Problem Chapter 6.2.1.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ b c(y-z) w_x +a c(z-x)w_y + a b (x -y) w_z= 0 \]

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]; 
 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*(z-x)*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];

\[ \text {\$Aborted} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*(z-x)*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),'build')),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;

\[ w \left ( x,y,z \right ) ={{\rm e}^{1/2\,{\it \_C2}\,{x}^{2}}}{{\rm e}^{{\it \_C1}\,x}}{{\rm e}^{1/2\,{\frac {b{y}^{2}{\it \_C2}}{a}}}}{{\rm e}^{{\frac {by{\it \_C1}}{a}}}}{\it \_C3}\,{\it \_C5}\,{\it \_C4}\,{{\rm e}^{1/2\,{\frac {c{z}^{2}{\it \_C2}}{a}}}}{{\rm e}^{{\frac {cz{\it \_C1}}{a}}}} \]

____________________________________________________________________________________

150.16 Problem 16

problem number 1191

Added April 13, 2019.

Problem Chapter 6.2.1.16, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ b c(b y-2 c z) w_x +a c(3 c z-a x) w_y + a b (2 a x -3 b y) w_z= 0 \]

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]; 
 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*(b*y-2*c*z)*D[w[x, y,z], x] +a*c*(3*c*z-a*x)*D[w[x, y,z], y] + a*b*(2*a*x -3*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];

\[ \text {\$Aborted} \]

Maple ✓ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*(b*y-2*c*z)*diff(w(x,y,z),x)+a*c*(3*c*z-a*x)*diff(w(x,y,z),y)+ a*b*(2*a*x-3*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;

\[ w \left ( x,y,z \right ) ={{\rm e}^{1/2\,{\it \_C2}\,{x}^{2}}}{{\rm e}^{{\it \_C1}\,x}}{{\rm e}^{1/2\,{\frac {{b}^{2}{\it \_C2}\,{y}^{2}}{{a}^{2}}}}}{{\rm e}^{2/3\,{\frac {by{\it \_C1}}{a}}}}{\it \_C3}\,{\it \_C5}\,{\it \_C4}\,{{\rm e}^{1/2\,{\frac {{c}^{2}{\it \_C2}\,{z}^{2}}{{a}^{2}}}}}{{\rm e}^{1/3\,{\frac {cz{\it \_C1}}{a}}}} \]

____________________________________________________________________________________

150.17 Problem 17

problem number 1192

Added April 13, 2019.

Problem Chapter 6.2.1.17, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ 2 b c(b y-c z) w_x -a c(4 a x-3 b y-c z) w_y + 3 a b (4 a x-b y-3 c z) w_z= 0 \]

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]; 
 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*b*c*(b*y-c*z)*D[w[x, y,z], x] -a*c*(4*a*x-3*b*y-c*z)*D[w[x, y,z], y] + 3*a*b*(4*a*x-b*y-3*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];

\[ \text {\$Aborted} \]

Maple ✗ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*b*c*(b*y-c*z)*diff(w(x,y,z),x)-a*c*(4*a*x-3*b*y-c*z)*diff(w(x,y,z),y)+ 3*a*b*(4*a*x-b*y-3*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;

\[ \text { sol=() } \]

____________________________________________________________________________________

150.18 Problem 18

problem number 1193

Added April 13, 2019.

Problem Chapter 6.2.1.18, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x+y-z) w_x -(x+a y-z) w_y + (a-1) (y-x) w_z= 0 \]

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]; 
 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+y-z)*D[w[x, y,z], x] -(x+a*y-z)*D[w[x, y,z], y] + (a-1)*(y-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];

\[ \text {\$Aborted} \]

Maple ✗ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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+y-z)*diff(w(x,y,z),x)-(x+a*y-z)*diff(w(x,y,z),y)+ (a-1)*(y-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;

\[ \text { sol=() } \]

____________________________________________________________________________________

150.19 Problem 19

problem number 1194

Added April 13, 2019.

Problem Chapter 6.2.1.19, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ 2 b c (3 a x-2 b y+c z) w_x -2 a c(2 a x-5 b y+3 c z) w_y + a b(2 a x-6 b y+11 c z) w_z= 0 \]

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]; 
 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*b*c*(3*a*x-2*b*y+c*z)*D[w[x, y,z], x] -2*a*c(2*a*x-5*b*y+3*c*z)*D[w[x, y,z], y] + a*b(2*a*x-6*b*y+11*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];

\[ \text {Failed} \]

Maple ✗ 

 
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,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*b*c*(3*a*x-2*b*y+c*z)*diff(w(x,y,z),x)-2*a*c*(2*a*x-5*b*y+3*c*z)*diff(w(x,y,z),y)+ a*b*(2*a*x-6*b*y+11*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;

\[ \text { sol=() } \]

____________________________________________________________________________________

150.20 Problem 20

problem number 1195

Added April 13, 2019.

Problem Chapter 6.2.1.20, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (A x+c y+b z) w_x +(c x+ B y+a z) w_y + (b x + a y + C z) w_z= 0 \]

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 =  (A*x+c*y+b*z)*D[w[x, y,z], x] +(c*x+B*y+a*z)*D[w[x, y,z], y] +(b*x+a*y+C1*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*x+c*y+b*z)*diff(w(x,y,z),x)+(c*x+B*y+a*z)*diff(w(x,y,z),y)+ (b*x+a*y+C1*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;

\[ \text { sol=() } \]

____________________________________________________________________________________

150.21 Problem 21

problem number 1196

Added April 13, 2019.

Problem Chapter 6.2.1.21, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a_1 x+b_1 y+c_1 z+d_1) w_x +(a_2 x+b_2 y+c_2 z+d_2) w_y + (a_3 x+b_3 y+c_3 z+d_3) w_z= 0 \]

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 =  (a1*x+b1*y+c1*z+d1)*D[w[x, y,z], x] +(a2*x+b2*y+c2*z+d2)*D[w[x, y,z], y] +(a3*x+b3*y+c3*z+d3)*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 :=   (a1*x+b1*y+c1*z+d1)*diff(w(x,y,z),x)+(a2*x+b2*y+c2*z+d2)*diff(w(x,y,z),y)+ (a3*x+b3*y+c3*z+d3)*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;

\[ \text { sol=() } \]

[next] [prev] [prev-tail] [front] [up]