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 $w(x,y)$

$$w_x+(a{e}^{\lambda x}y+b{x}^n)w_y=cw+k{e}^{\gamma x}$$

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);
 

$$w(x,y)=(\frac{k{\mathrm{e}}^{x(\gamma -c)}}{\gamma -c}+\mathit{\_}\mathit{F}\mathit{1}(-b\int x^n{\mathrm{e}}^{-\frac{a{\mathrm{e}}^{\lambda x}}{\lambda }}\mathrm{d}x+y{\mathrm{e}}^{-\frac{a{\mathrm{e}}^{\lambda x}}{\lambda }})){\mathrm{e}}^{cx}$$

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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 $w(x,y)$

$$w_x+(a{e}^{\lambda x}y+b{e}^{\beta x})w_y=cw+k{e}^{\gamma x}$$

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];
 

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

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);
 

$$w(x,y)=-\frac{{\mathrm{e}}^{cx}}{-\gamma +c}((\gamma -c)\mathit{\_}\mathit{F}\mathit{1}(-b\int {\mathrm{e}}^{\frac{\beta x\lambda -a{\mathrm{e}}^{\lambda x}}{\lambda }}\mathrm{d}x+y{\mathrm{e}}^{-\frac{a{\mathrm{e}}^{\lambda x}}{\lambda }})+k{\mathrm{e}}^{x(\gamma -c)})$$

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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 $w(x,y)$

$$w_x+(a{e}^{\lambda x}y+b{e}^{\beta x})w_y=cw+k{x}^n$$

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];
 

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

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);
 

$$w(x,y)=\frac{{\mathrm{e}}^{cx}}{c(n+1)}(c(n+1)\mathit{\_}\mathit{F}\mathit{1}(-b\int {\mathrm{e}}^{\frac{\beta x\lambda -a{\mathrm{e}}^{\lambda x}}{\lambda }}\mathrm{d}x+y{\mathrm{e}}^{-\frac{a{\mathrm{e}}^{\lambda x}}{\lambda }})+k{x}^n{\left(cx\right)}^{-n/2}{\mathrm{e}}^{-1/2cx}\mathrm{WhittakerM}(n/2,n/2+1/2,cx))$$

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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 $w(x,y)$

$$w_x+(a{e}^{\lambda y}+b{x}^k)w_y=cw+k{e}^{\gamma x}$$

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];
 

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

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);
 

$$w(x,y)=-\frac{{\mathrm{e}}^{cx}}{-\gamma +c}((\gamma -c)\mathit{\_}\mathit{F}\mathit{1}\left(\frac{1}{\lambda b(2k^2+7k+6)}(a{(-\frac{x^{k+1}\lambda b}{k+1})}^{\frac{-k-2}{2k+2}}{x}^{-k}{\mathrm{e}}^{\frac{x^{k+1}\lambda b}{2k+2}}(k+1){(k+2)}^2\mathrm{WhittakerM}(\frac{k+2}{2k+2},\frac{2k+3}{2k+2},-\frac{x^{k+1}\lambda b}{k+1})-a{(-\frac{x^{k+1}\lambda b}{k+1})}^{\frac{-k-2}{2k+2}}((-k-2)x^{-k}+bx\lambda ){(k+1)}^2{\mathrm{e}}^{\frac{x^{k+1}\lambda b}{2k+2}}\mathrm{WhittakerM}(-\frac{k}{2k+2},\frac{2k+3}{2k+2},-\frac{x^{k+1}\lambda b}{k+1})-2{\mathrm{e}}^{\frac{(x^{k+1}b-y(k+1))\lambda }{k+1}}(3/2+k)b(k+2))\right)+k{\mathrm{e}}^{x(\gamma -c)})$$

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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 $w(x,y)$

$$x{w}_x+y{w}_y=ax{e}^{\lambda x+\mu y}w+b{e}^{\nu x}$$

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];
 

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

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);
 

$$w(x,y)=({\int }^x\frac{k}{\mathit{\_}\mathit{a}}{\mathrm{e}}^{-\frac{1}{\lambda x+\mu y}(ax{\mathrm{e}}^{\frac{\mu y\mathit{\_}\mathit{a}}{x}+\lambda \mathit{\_}\mathit{a}}-\nu \mathit{\_}\mathit{a}(\lambda x+\mu y))}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{y}{x}\right)){\mathrm{e}}^{a{\mathrm{e}}^{\lambda x+\mu y}{(\frac{\mu y}{x}+\lambda )}^{-1}}$$

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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 $w(x,y)$

$$x{w}_x+y{w}_y=(ay{e}^{\lambda x}+bx{e}^{\mu y})w+c{e}^{\nu x}$$

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];
 

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

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);
 

$$w(x,y)=({\int }^x\frac{c}{\mathit{\_}\mathit{a}}{\mathrm{e}}^{-\frac{x}{\lambda \mu y}(\frac{{\mathrm{e}}^{\lambda \mathit{\_}\mathit{a}}{y}^{2}a\mu }{x^2}-\frac{\nu \mathit{\_}\mathit{a}\mu y\lambda }{x}+{\mathrm{e}}^{\frac{\mu y\mathit{\_}\mathit{a}}{x}}b\lambda )}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{y}{x}\right)){\mathrm{e}}^{\frac{x}{\lambda \mu y}(\frac{a{\mathrm{e}}^{\lambda x}{y}^2\mu }{x^2}+{\mathrm{e}}^{\mu y}b\lambda )}$$

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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];
 

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

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);
 

$$w(x,y)=({\int }^x\frac{c}{a}{\left({\left(\frac{b(k+1){\mathrm{e}}^{\mathit{\_}\mathit{b}\lambda }-{\mathrm{e}}^{\lambda x}b(k+1)+y^{k}ya\lambda }{a\lambda }\right)}^{{(k+1)}^{-1}}\right)}^{-k}{\mathrm{e}}^{\frac{1}{a}(\mathit{\_}\mathit{b}a\beta -\int {\left({\left(\frac{b(k+1){\mathrm{e}}^{\mathit{\_}\mathit{b}\lambda }-{\mathrm{e}}^{\lambda x}b(k+1)+y^{k}ya\lambda }{a\lambda }\right)}^{{(k+1)}^{-1}}\right)}^{-k}\mathrm{d}\mathit{\_}\mathit{b})}d\mathit{\_}\mathit{b}+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{-{\mathrm{e}}^{\lambda x}b(k+1)+y^{k}ya\lambda }{a\lambda }\right)){\mathrm{e}}^{{\int }^x\frac{1}{a}{\left({\left(\frac{b(k+1){\mathrm{e}}^{\lambda \mathit{\_}\mathit{a}}-{\mathrm{e}}^{\lambda x}b(k+1)+y^{k}ya\lambda }{a\lambda }\right)}^{{(k+1)}^{-1}}\right)}^{-k}d\mathit{\_}\mathit{a}}$$

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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 $w(x,y)$

$$a{e}^{\lambda x}{w}_x+by{w}_y=w+c{e}^{\lambda x}$$

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];
 

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

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);
 

$$w(x,y)=\frac{1}{a\lambda }(\mathit{\_}\mathit{F}\mathit{1}\left(y{\mathrm{e}}^{\frac{b{\mathrm{e}}^{-\lambda x}}{a\lambda }}\right)a\lambda +c\mathrm{expIntegral}(1,-\frac{{\mathrm{e}}^{-\lambda x}}{a\lambda })){\mathrm{e}}^{-\frac{{\mathrm{e}}^{-\lambda x}}{a\lambda }}$$

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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];
 

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

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);
 

$$w(x,y)=({\int }^x\frac{c(k+1)}{-x^{k+1}\lambda b+{\mathit{\_}\mathit{b}}^{k+1}\lambda b+{\mathrm{e}}^{y\lambda }a(k+1)}{\mathrm{e}}^{\frac{1}{\lambda b}((-1-k)\int \frac{\lambda b}{-x^{k+1}\lambda b+{\mathit{\_}\mathit{b}}^{k+1}\lambda b+{\mathrm{e}}^{y\lambda }a(k+1)}\mathrm{d}\mathit{\_}\mathit{b}+b\mathit{\_}\mathit{b}\lambda \beta )}d\mathit{\_}\mathit{b}+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{-x^{k+1}\lambda b+{\mathrm{e}}^{y\lambda }a(k+1)}{(k+1)\lambda b}\right)){\mathrm{e}}^{{\int }^x\frac{k+1}{-x^{k+1}\lambda b+{\mathit{\_}\mathit{a}}^{k+1}\lambda b+{\mathrm{e}}^{y\lambda }a(k+1)}d\mathit{\_}\mathit{a}}$$

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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];
 

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

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);
 

$$w(x,y)=({\int }^x\frac{c{\mathit{\_}\mathit{a}}^k\beta }{\lambda b}{(\frac{{\mathrm{e}}^{y\lambda }a\beta -{\mathrm{e}}^{\beta x}b\lambda }{\lambda b}+{\mathrm{e}}^{\beta \mathit{\_}\mathit{a}})}^{\frac{-{\mathrm{e}}^{y\lambda }a\beta +{\mathrm{e}}^{\beta x}b\lambda +1}{{\mathrm{e}}^{y\lambda }a\beta -{\mathrm{e}}^{\beta x}b\lambda }}{\left({\mathrm{e}}^{\beta \mathit{\_}\mathit{a}}\right)}^{-{({\mathrm{e}}^{y\lambda }a\beta -{\mathrm{e}}^{\beta x}b\lambda )}^{-1}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{{\mathrm{e}}^{y\lambda }a\beta -{\mathrm{e}}^{\beta x}b\lambda }{b\beta \lambda }\right)){\left(\frac{{\mathrm{e}}^{y\lambda }a\beta }{\lambda b}\right)}^{-{({\mathrm{e}}^{y\lambda }a\beta -{\mathrm{e}}^{\beta x}b\lambda )}^{-1}}{\left({\mathrm{e}}^{\beta x}\right)}^{{({\mathrm{e}}^{y\lambda }a\beta -{\mathrm{e}}^{\beta x}b\lambda )}^{-1}}$$