139 HFOPDE, chapter 5.6.2

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139.1 Problem 1

problem number 1105

Added April 11, 2019.

Problem Chapter 5.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + k \cos (\lambda x+\mu y) \]

Mathematica ✓

ClearAll[w, x, y, 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]; 
 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], x] + b*D[w[x, y], y] == c*w[x,y]+ k*Cos[lambda*x+mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \frac {k ((a \lambda +b \mu ) \sin (\lambda x+\mu y)-c \cos (\lambda x+\mu y))}{(a \lambda +b \mu )^2+c^2}+e^{\frac {c x}{a}} c_1\left (y-\frac {b x}{a}\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,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),x)+ b*diff(w(x,y),y) = c*w(x,y)+ k*cos(lambda*x+mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) ={\frac {1}{{a}^{2}{\lambda }^{2}+2\,ab\lambda \,\mu +{\mu }^{2}{b}^{2}+{c}^{2}} \left ( \left ( {a}^{2}{\lambda }^{2}+2\,ab\lambda \,\mu +{\mu }^{2}{b}^{2}+{c}^{2} \right ) {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) +k{{\rm e}^{-{\frac {cx}{a}}}} \left ( \left ( a\lambda +b\mu \right ) \sin \left ( \lambda \,x+\mu \,y \right ) -c\cos \left ( \lambda \,x+\mu \,y \right ) \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}} \]

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139.2 Problem 2

problem number 1106

Added April 11, 2019.

Problem Chapter 5.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = w + c_1 \cos ^k(\lambda x) + c_2 \cos ^n(\beta y) \]

Mathematica ✓

ClearAll[w, x, y, 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]; 
 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], x] + b*D[w[x, y], y] == w[x,y]+ c1*Cos[lambda*x]^k + c2*Cos[beta*y]^n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \frac {\text {c1} 2^{-k} (1+i b \beta n) \left (e^{-i \lambda x}+e^{i \lambda x}\right )^k \left (1+e^{2 i \lambda x}\right )^{-k} \text {Hypergeometric2F1}\left (-k,-\frac {k}{2}+\frac {i}{2 a \lambda },\frac {i}{2 a \lambda }-\frac {k}{2}+1,-e^{2 i \lambda x}\right )+\text {c2} 2^n (1+i a k \lambda ) \cos ^n(\beta y) (\cosh (n \log (2))-\sinh (n \log (2))) (i \sin (2 \beta y)+\cos (2 \beta y)+1)^{-n} \text {Hypergeometric2F1}\left (-\frac {n}{2}+\frac {i}{2 b \beta },-n,\frac {i}{2 b \beta }-\frac {n}{2}+1,-\cos (2 \beta y)-i \sin (2 \beta y)\right )+e^{\frac {x}{a}} (a k \lambda -i) (b \beta n-i) c_1\left (y-\frac {b x}{a}\right )}{(a k \lambda -i) (b \beta n-i)}\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,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),x)+ b*diff(w(x,y),y) = w(x,y)+ c1*cos(lambda*x)^k + c2*cos(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) ={{\rm e}^{{\frac {x}{a}}}} \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) +\int ^{x}\!{\frac {1}{a} \left ( {\it c2}\, \left ( \cos \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}+{\it c1}\, \left ( \cos \left ( \lambda \,{\it \_a} \right ) \right ) ^{k} \right ) {{\rm e}^{-{\frac {{\it \_a}}{a}}}}}{d{\it \_a}} \right ) \]

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139.3 Problem 3

problem number 1107

Added April 11, 2019.

Problem Chapter 5.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + \cos ^k(\lambda x) \cos ^n(\beta y) \]

Mathematica ✓

ClearAll[w, x, y, 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]; 
 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], x] + b*D[w[x, y], y] == c*w[x,y]+ Cos[lambda*x]^k * Cos[beta*y]^n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x \frac {e^{-\frac {c K[1]}{a}} \cos ^k(\lambda K[1]) \cos ^n\left (\beta \left (\frac {b (K[1]-x)}{a}+y\right )\right )}{a} \, dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,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),x)+ b*diff(w(x,y),y) = c*w(x,y)+ cos(lambda*x)^k *cos(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) ={{\rm e}^{{\frac {cx}{a}}}} \left ( \int ^{x}\!{\frac { \left ( \cos \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}}{a} \left ( \cos \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) \]

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139.4 Problem 4

problem number 1108

Added April 11, 2019.

Problem Chapter 5.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = c w + k \cos (\lambda x+\mu y) \]

Mathematica ✓

ClearAll[w, x, y, 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]; 
 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*D[w[x, y], x] + b*y*D[w[x, y], y] == c*w[x,y]+ k*Cos[lambda*x+mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to x^{\frac {c}{a}} \left (\int _1^x \frac {k K[1]^{-\frac {a+c}{a}} \cos \left (\mu y x^{-\frac {b}{a}} K[1]^{\frac {b}{a}}+\lambda K[1]\right )}{a} \, dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,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*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*w(x,y)+ k*cos(lambda*x+mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) ={x}^{{\frac {c}{a}}} \left ( {\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) +\int ^{x}\!{\frac {k}{a}\cos \left ( \lambda \,{\it \_a}+\mu \,y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) {{\it \_a}}^{{\frac {-a-c}{a}}}}{d{\it \_a}} \right ) \]

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139.5 Problem 5

problem number 1109

Added April 11, 2019.

Problem Chapter 5.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x \cos (\lambda x+\mu y) w + b \cos (\nu x) \]

Mathematica ✓

ClearAll[w, x, y, 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]; 
 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], x] + y*D[w[x, y], y] == a*x*Cos[lambda*x+mu*y]*w[x,y]+b*Cos[nu*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\frac {a x \sin (\lambda x+\mu y)}{\lambda x+\mu y}} \left (\int _1^x \frac {b \cos (\nu K[1]) \exp \left (-\frac {a x \sin \left (K[1] \left (\lambda +\frac {\mu y}{x}\right )\right )}{\lambda x+\mu y}\right )}{K[1]} \, dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,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),x)+ y*diff(w(x,y),y) =a*x*cos(lambda*x+mu*y)*w(x,y)+b*cos(nu*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) ={{\rm e}^{{a\sin \left ( \lambda \,x+\mu \,y \right ) \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}}} \left ( \int ^{x}\!{\frac {\cos \left ( \nu \,{\it \_a} \right ) b}{{\it \_a}}{{\rm e}^{-{a\sin \left ( {\frac {\mu \,y{\it \_a}}{x}}+\lambda \,{\it \_a} \right ) \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) \]

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139.6 Problem 6

problem number 1110

Added April 11, 2019.

Problem Chapter 5.6.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \cos ^n(\lambda x) w_x + b \cos ^m(\mu x) w_y = c \cos ^k(\nu x) w + p \cos ^s(\beta y) \]

Mathematica ✗

ClearAll[w, x, y, 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]; 
 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*Cos[lambda*x]^n*D[w[x, y], x] + b*Cos[mu*x]^m*D[w[x, y], y] == c*Cos[nu*x]^k*w[x,y]+p*Cos[beta*y]^s; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \]

Maple ✓

 
unassign('w,x,y,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*cos(lambda*x)^n*diff(w(x,y),x)+ b*cos(mu*x)^m*diff(w(x,y),y) =c*cos(nu*x)^k*w(x,y)+p*cos(beta*y)^s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) ={{\rm e}^{\int \!{\frac { \left ( \cos \left ( \nu \,x \right ) \right ) ^{k}c \left ( \cos \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}x}} \left ( \int ^{x}\!{\frac {p \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}}{a} \left ( \cos \left ( {\frac {\beta \, \left ( b\int \! \left ( \cos \left ( {\it \_f}\,\mu \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}\,{\rm d}{\it \_f}+ya-b\int \! \left ( \cos \left ( \mu \,x \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x \right ) }{a}} \right ) \right ) ^{s}{{\rm e}^{-{\frac {c\int \! \left ( \cos \left ( \nu \,{\it \_f} \right ) \right ) ^{k} \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}\,{\rm d}{\it \_f}}{a}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {ya-b\int \! \left ( \cos \left ( \mu \,x \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x}{a}} \right ) \right ) \]