138 HFOPDE, chapter 5.6.1

____________________________________________________________________________________

138.1 Problem 1

problem number 1098

Added April 8, 2019.

Problem Chapter 5.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Sin[lambda*x+mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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*sin(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;
 

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

____________________________________________________________________________________

138.2 Problem 2

problem number 1099

Added April 8, 2019.

Problem Chapter 5.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[ \left \{\left \{w(x,y)\to e^{\frac {x}{a}} c_1\left (y-\frac {b x}{a}\right )-i \left (\frac {\text {c1} \left (-1+e^{2 i \lambda x}\right ) \sin ^k(\lambda x) \, _2F_1\left (1,\frac {1}{2} \left (k+\frac {i}{a \lambda }+2\right );\frac {1}{2} \left (-k+\frac {i}{a \lambda }+2\right );e^{2 i \lambda x}\right )}{a k \lambda -i}+\frac {\text {c2} \left (-1+e^{2 i \beta y}\right ) \sin ^n(\beta y) \, _2F_1\left (1,\frac {1}{2} \left (n+\frac {i}{b \beta }+2\right );\frac {1}{2} \left (-n+\frac {i}{b \beta }+2\right );e^{2 i \beta y}\right )}{b \beta n-i}\right )\right \}\right \} \]

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) = w(x,y)+c1*sin(lambda*x)^k+c2*sin(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;
 

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

____________________________________________________________________________________

138.3 Problem 3

problem number 1100

Added April 8, 2019.

Problem Chapter 5.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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)+sin(lambda*x)^k*sin(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;
 

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

____________________________________________________________________________________

138.4 Problem 4

problem number 1101

Added April 8, 2019.

Problem Chapter 5.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Sin[lambda*x+beta*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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*sin(lambda*x+beta*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;
 

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

____________________________________________________________________________________

138.5 Problem 5

problem number 1102

Added April 8, 2019.

Problem Chapter 5.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Sin[lambda*x+beta*y]*w[x,y]+ b*Sin[nu*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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)+ b*diff(w(x,y),y) = a*x*sin(lambda*x+beta*y)*w(x,y)+ b*sin(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;
 

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

____________________________________________________________________________________

138.6 Problem 6

problem number 1103

Added April 8, 2019.

Problem Chapter 5.6.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Sin[lambda*x]^n*D[w[x, y], x] + b*Sin[mu*x]^m*D[w[x, y], y] == c*Sin[nu*x]^k*w[x,y]+ p*Sin[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*sin(lambda*x)^n*diff(w(x,y),x)+ b*sin(mu*x)^m*diff(w(x,y),y) = c*sin(nu*x)^k*w(x,y)+ p*sin(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;
 

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

____________________________________________________________________________________

138.7 Problem 7

problem number 1104

Added April 8, 2019.

Problem Chapter 5.6.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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 = a*Sin[lambda*x]^n*D[w[x, y], x] + b*Sin[mu*x]^m*D[w[x, y], y] == c*Sin[nu*y]^k*w[x,y]+ p*Sin[beta*x]^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*sin(lambda*x)^n*diff(w(x,y),x)+ b*sin(mu*x)^m*diff(w(x,y),y) = c*sin(nu*y)^k*w(x,y)+ p*sin(beta*x)^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;
 

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