129 HFOPDE, chapter 5.3.1

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129.1 Problem 1

problem number 1040

Added April 1, 2019.

Problem Chapter 5.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*exp(lambda*x)+s*exp(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);
 

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

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129.2 Problem 2

problem number 1041

Added April 1, 2019.

Problem Chapter 5.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]; 
 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*Exp[alpha*x+beta*y]*w[x,y] + k*Exp[gamma*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*exp(alpha*x+beta*y)*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);
 

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

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129.3 Problem 3

problem number 1042

Added April 1, 2019.

Problem Chapter 5.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda x} w_x + b e^{\beta x} w_y = c e^{\gamma y} w+ s e^{\mu x+\delta y} \]

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]; 
 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*Exp[beta*x]*D[w[x, y], y] == c*Exp[gamma*y]*w[x,y] + s*Exp[mu*x+delta*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \exp \left (\int _1^x \frac {c \exp \left (-\frac {b \gamma \left (e^{x (\beta -\lambda )}-e^{(\beta -\lambda ) K[1]}\right )}{a (\beta -\lambda )}-\lambda K[1]+\gamma y\right )}{a} \, dK[1]\right ) \left (\int _1^x \frac {s \exp \left (-\text {Integrate}\left [\frac {c \exp \left (-\frac {b \gamma \left (e^{x (\beta -\lambda )}-e^{(\beta -\lambda ) K[1]}\right )}{a (\beta -\lambda )}-\lambda K[1]+\gamma y\right )}{a},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]-\frac {b \delta \left (e^{x (\beta -\lambda )}-e^{(\beta -\lambda ) K[2]}\right )}{a (\beta -\lambda )}+(\mu -\lambda ) K[2]+\delta y\right )}{a} \, dK[2]+c_1\left (\frac {b e^{x (\beta -\lambda )}}{a (\lambda -\beta )}+y\right )\right )\right \}\right \} \]

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

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

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129.4 Problem 4

problem number 1043

Added April 1, 2019.

Problem Chapter 5.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[ \text {Failed} \]

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

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

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129.5 Problem 5

problem number 1044

Added April 1, 2019.

Problem Chapter 5.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[ \text {Failed} \]

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

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

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129.6 Problem 6

problem number 1045

Added April 1, 2019.

Problem Chapter 5.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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[beta*x]*D[w[x, y], x] + b*Exp[gamma*x+lambda*y]*D[w[x, y], y] == c*Exp[sigma*y]*w[x,y]+k*Exp[mu*x+delta*y]+d; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \]

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

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

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129.7 Problem 7

problem number 1046

Added April 1, 2019.

Problem Chapter 5.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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129.8 Problem 8

problem number 1047

Added April 1, 2019.

Problem Chapter 5.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*x^(beta*x)*D[w[x, y], y] == c*Exp[gamma*x]*w[x,y]+s; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \]

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

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