130 HFOPDE, chapter 5.3.2

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130.1 Problem 1

problem number 1048

Added April 2, 2019.

Problem Chapter 5.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a e^{\lambda x} y+ b x^n) w_y = c w + k e^{\gamma x} \]

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]; 
 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], x] + (a*Exp[lambda*x]*y+b*x^n)*D[w[x, y], y] == c*w[x,y]+k*Exp[gamma*x]; 
 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'); 
pde :=  diff(w(x,y),x)+ (a*exp(lambda*x)*y+b*x^n)*diff(w(x,y),y) = c*w(x,y)+k*exp(gamma*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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130.2 Problem 2

problem number 1049

Added April 2, 2019.

Problem Chapter 5.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a e^{\lambda x} y+ b e^{\beta x}) w_y = c w + k e^{\gamma x} \]

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]; 
 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], x] + (a*Exp[lambda*x]*y+b*Exp[beta*x])*D[w[x, y], y] == c*w[x,y]+k*Exp[gamma*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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'); 
pde :=  diff(w(x,y),x)+ (a*exp(lambda*x)*y+b*exp(beta*x))*diff(w(x,y),y) = c*w(x,y)+k*exp(gamma*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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130.3 Problem 3

problem number 1050

Added April 2, 2019.

Problem Chapter 5.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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'); 
pde :=  diff(w(x,y),x)+ (a*exp(lambda*x)*y+b*exp(beta*x))*diff(w(x,y),y) = c*w(x,y)+k*x^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

\[ w \left ( x,y \right ) ={\frac {{{\rm e}^{cx}}}{c \left ( n+1 \right ) } \left ( c \left ( n+1 \right ) {\it \_F1} \left ( -b\int \!{{\rm e}^{{\frac {\beta \,x\lambda -a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {a{{\rm e}^{\lambda \,x}}}{\lambda }}}} \right ) +k{x}^{n} \left ( cx \right ) ^{-n/2}{{\rm e}^{-1/2\,cx}} \WhittakerM \left ( n/2,n/2+1/2,cx \right ) \right ) } \]

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130.4 Problem 4

problem number 1051

Added April 2, 2019.

Problem Chapter 5.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a e^{\lambda y} + b x^k) w_y = c w + k e^{\gamma x} \]

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]; 
 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], x] + (a*Exp[lambda*y]+b*x^k)*D[w[x, y], y] == c*w[x,y]+k*Exp[gamma*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{c x} \left (c_1\left (\frac {a \lambda x \left (-\frac {b \lambda x^{k+1}}{k+1}\right )^{-\frac {1}{k+1}} \text {Gamma}\left (\frac {1}{k+1},-\frac {b \lambda x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\lambda \left (-b x^{k+1}+k y+y\right )}{k+1}}}{a b k (k+1) \lambda ^2}\right )-\frac {k e^{x (\gamma -c)}}{c-\gamma }\right )\right \}\right \} \]

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'); 
pde :=  diff(w(x,y),x)+ (a*exp(lambda*y)+b*x^k)*diff(w(x,y),y) = c*w(x,y)+k*exp(gamma*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

\[ w \left ( x,y \right ) =-{\frac {{{\rm e}^{cx}}}{-\gamma +c} \left ( \left ( \gamma -c \right ) {\it \_F1} \left ( {\frac {1}{\lambda \,b \left ( 2\,{k}^{2}+7\,k+6 \right ) } \left ( a \left ( -{\frac {{x}^{k+1}\lambda \,b}{k+1}} \right ) ^{{\frac {-k-2}{2\,k+2}}}{x}^{-k}{{\rm e}^{{\frac {{x}^{k+1}\lambda \,b}{2\,k+2}}}} \left ( k+1 \right ) \left ( k+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+2}{2\,k+2}},{\frac {2\,k+3}{2\,k+2}},-{\frac {{x}^{k+1}\lambda \,b}{k+1}} \right ) -a \left ( -{\frac {{x}^{k+1}\lambda \,b}{k+1}} \right ) ^{{\frac {-k-2}{2\,k+2}}} \left ( \left ( -k-2 \right ) {x}^{-k}+bx\lambda \right ) \left ( k+1 \right ) ^{2}{{\rm e}^{{\frac {{x}^{k+1}\lambda \,b}{2\,k+2}}}} \WhittakerM \left ( -{\frac {k}{2\,k+2}},{\frac {2\,k+3}{2\,k+2}},-{\frac {{x}^{k+1}\lambda \,b}{k+1}} \right ) -2\,{{\rm e}^{{\frac { \left ( {x}^{k+1}b-y \left ( k+1 \right ) \right ) \lambda }{k+1}}}} \left ( 3/2+k \right ) b \left ( k+2 \right ) \right ) } \right ) +k{{\rm e}^{x \left ( \gamma -c \right ) }} \right ) } \]

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130.5 Problem 5

problem number 1052

Added April 2, 2019.

Problem Chapter 5.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x e^{\lambda x+\mu y} w + b e^{\nu x} \]

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

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

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'); 
pde :=  x* diff(w(x,y),x)+ y*diff(w(x,y),y) =  a*x*exp(lambda*x+mu*y)*w(x,y)+k*exp(nu*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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130.6 Problem 6

problem number 1053

Added April 2, 2019.

Problem Chapter 5.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = (a y e^{\lambda x}+ b x e^{\mu y}) w + c e^{\nu x} \]

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]; 
 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*y*Exp[lambda*x]+b*x*Exp[mu*y])*w[x,y]+c*Exp[nu*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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'); 
pde :=  x* diff(w(x,y),x)+ y*diff(w(x,y),y) = (a*y*exp(lambda*x)+b*x*exp(mu*y))*w(x,y)+c*exp(nu*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {c}{{\it \_a}}{{\rm e}^{-{\frac {x}{\lambda \,\mu \,y} \left ( {\frac {{{\rm e}^{\lambda \,{\it \_a}}}{y}^{2}a\mu }{{x}^{2}}}-{\frac {\nu \,{\it \_a}\,\mu \,y\lambda }{x}}+{{\rm e}^{{\frac {\mu \,y{\it \_a}}{x}}}}b\lambda \right ) }}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) {{\rm e}^{{\frac {x}{\lambda \,\mu \,y} \left ( {\frac {a{{\rm e}^{\lambda \,x}}{y}^{2}\mu }{{x}^{2}}}+{{\rm e}^{\mu \,y}}b\lambda \right ) }}} \]

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130.7 Problem 7

problem number 1054

Added April 2, 2019.

Problem Chapter 5.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b e^{\lambda x} w_y = w + c e^{\beta x} \]

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]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*y^k*D[w[x, y], x] + b*Exp[lambda*x]*D[w[x, y], y] == w[x,y]+c*Exp[beta*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \exp \left (-\frac {(k+1) \left (\left (y^{k+1}\right )^{\frac {1}{k+1}}\right )^{-k} \left (\frac {a \lambda y^{k+1} e^{-\lambda x}}{b k+b}\right )^{\frac {k}{k+1}} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {k}{k+1},\frac {k}{k+1}+1,1-\frac {a \lambda y^{k+1} e^{-\lambda x}}{b k+b}\right )}{a k \lambda }\right ) \left (\int _1^x \frac {c \left (\left (y^{k+1}-\frac {b (k+1) \left (e^{\lambda x}-e^{\lambda K[1]}\right )}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k} \exp \left (\frac {(k+1) \left (\frac {a \lambda y^{k+1} e^{-\lambda K[1]}}{b k+b}-e^{\lambda (x-K[1])}+1\right )^{\frac {k}{k+1}} \left (\left (y^{k+1}-\frac {b (k+1) \left (e^{\lambda x}-e^{\lambda K[1]}\right )}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {k}{k+1},\frac {k}{k+1}+1,\frac {e^{-\lambda K[1]} \left (b (k+1) e^{\lambda x}-a \lambda y^{k+1}\right )}{b (k+1)}\right )}{a k \lambda }+\beta K[1]\right )}{a} \, dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b e^{\lambda x}}{a \lambda }\right )\right )\right \}\right \} \]

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'); 
pde :=  a*y^k* diff(w(x,y),x)+ b*exp(lambda*x)*diff(w(x,y),y) = w(x,y)+c*exp(beta*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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130.8 Problem 8

problem number 1055

Added April 2, 2019.

Problem Chapter 5.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda x} w_x + b y w_y = w + c e^{\lambda x} \]

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]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*Exp[lambda*x]*D[w[x, y], x] + b*y*D[w[x, y], y] == w[x,y]+c*Exp[lambda*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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'); 
pde :=  a*exp(lambda*x)* diff(w(x,y),x)+ b*y*diff(w(x,y),y) = w(x,y)+c*exp(lambda*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

\[ w \left ( x,y \right ) ={\frac {1}{a\lambda } \left ( {\it \_F1} \left ( y{{\rm e}^{{\frac {b{{\rm e}^{-\lambda \,x}}}{a\lambda }}}} \right ) a\lambda +c\Ei \left ( 1,-{\frac {{{\rm e}^{-\lambda \,x}}}{a\lambda }} \right ) \right ) {{\rm e}^{-{\frac {{{\rm e}^{-\lambda \,x}}}{a\lambda }}}}} \]

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130.9 Problem 9

problem number 1056

Added April 2, 2019.

Problem Chapter 5.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda y} w_x + b x^k w_y = w + c e^{\beta x} \]

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]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*Exp[lambda*y]*D[w[x, y], x] + b*x^k*D[w[x, y], y] == w[x,y]+c*Exp[beta*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \exp \left (\frac {(k+1) x \text {Hypergeometric2F1}\left (1,\frac {1}{k+1},\frac {1}{k+1}+1,\frac {b \lambda x^{k+1}}{b \lambda x^{k+1}-a (k+1) e^{\lambda y}}\right )}{a (k+1) e^{\lambda y}-b \lambda x^{k+1}}\right ) \left (\int _1^x \frac {c (k+1) \exp \left (\frac {K[1] \left (-(k+1) \text {Hypergeometric2F1}\left (1,\frac {1}{k+1},\frac {1}{k+1}+1,\frac {b \lambda K[1]^{k+1}}{b \lambda x^{k+1}-a (k+1) e^{\lambda y}}\right )+a \beta (k+1) e^{\lambda y}-b \beta \lambda x^{k+1}\right )}{a (k+1) e^{\lambda y}-b \lambda x^{k+1}}\right )}{b \lambda \left (K[1]^{k+1}-x^{k+1}\right )+a (k+1) e^{\lambda y}} \, dK[1]+c_1\left (\frac {e^{\lambda y}}{\lambda }-\frac {b x^{k+1}}{a k+a}\right )\right )\right \}\right \} \]

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'); 
pde :=  a*exp(lambda*y)* diff(w(x,y),x)+ b*x^k*diff(w(x,y),y) = w(x,y)+c*exp(beta*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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130.10 Problem 10

problem number 1057

Added April 2, 2019.

Problem Chapter 5.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda y} w_x + b e^{\beta x} w_y = w + c x^k \]

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]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*Exp[lambda*y]*D[w[x, y], x] + b*Exp[beta*x]*D[w[x, y], y] == w[x,y]+c*x^k; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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'); 
pde :=  a*exp(lambda*y)* diff(w(x,y),x)+ b*exp(beta*x)*diff(w(x,y),y) = w(x,y)+c*x^k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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