7.5.10 4.4

7.5.10.1 [1267] Problem 1

problem number 1267

Added April 3, 2019.

Problem Chapter 5.4.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{w}_x+b{w}_y=cw+{\coth }^k(\lambda x){\coth }^n(\beta y)$$

Mathematica ✓

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

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

Maple ✓

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

$$w(x,y)=({\int }_{}^x\frac{({\coth }^k\left(\mathit{\text{\_a}}\lambda \right))({\coth }^n\left(\frac{(ay-(-\mathit{\text{\_a}}+x)b)\beta }{a}\right)){\mathrm{e}}^{-\frac{\mathit{\text{\_a}}c}{a}}}{a}d\mathit{\text{\_a}}+\mathit{\text{\_F1}}\left(\frac{ay-bx}{a}\right)){\mathrm{e}}^{\frac{cx}{a}}$$

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7.5.10.2 [1268] Problem 2

problem number 1268

Added April 3, 2019.

Problem Chapter 5.4.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{w}_x+b{w}_y=c{\coth }^k(\lambda x)w+s{\coth }^n(\beta x)$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*Coth[lambda*x]^k*w[x,y]+ s*Coth[beta*x]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to \exp \left(\frac{c{\coth }^{k+1}(\lambda x){}_2{F}_1(1,\frac{k+1}{2};\frac{k+3}{2};{\coth }^2(\lambda x))}{ak\lambda +a\lambda }\right)({\int }_1^x\frac{\exp (-\frac{c{\coth }^{k+1}(\lambda K[1]){}_2{F}_1(1,\frac{k+1}{2};\frac{k+3}{2};{\coth }^2(\lambda K[1]))}{a\lambda +ak\lambda })s{\coth }^n(\beta K[1])}{a}dK[1]+c_1(y-\frac{bx}{a}))\}\right\}$$

Maple ✓

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

$$w(x,y)=(\int \frac{s({\coth }^n\left(\beta x\right)){\mathrm{e}}^{-\frac{c(\int ({\coth }^k\left(\lambda x\right))dx)}{a}}}{a}dx+\mathit{\text{\_F1}}\left(\frac{ay-bx}{a}\right)){\mathrm{e}}^{\int \frac{c({\coth }^k\left(\lambda x\right))}{a}dx}$$

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7.5.10.3 [1269] Problem 3

problem number 1269

Added April 3, 2019.

Problem Chapter 5.4.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{w}_x+b{w}_y=(c_1{\coth }^{n_1}({\lambda }_{1}x)+c_2{\coth }^{n_2}({\lambda }_{2}y))w+s_1{\coth }^{k_1}({\beta }_{1}x)+s_2{\coth }^{k_2}({\beta }_{2}y)$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (c1*Coth[lambda1*x]^n1 + c2*Coth[lambda2*y]^n2)*w[x,y] + s1*Coth[beta1*x]^k1+ s2*Coth[beta2*y]^k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$Aborted

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c1*coth(lambda1*x)^n1 + c2*coth(lambda2*y)^n2)*w(x,y) + s1*coth(beta1*x)^k1+ s2*coth(beta2*y)^k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=({\int }_{}^x\frac{(\mathit{s}\mathit{1}({\coth }^{\mathit{k}\mathit{1}}\left(\mathit{\text{\_b}}\beta 1\right))+\mathit{s}\mathit{2}({\coth }^{\mathit{k}\mathit{2}}\left(\frac{(ay-(-\mathit{\text{\_b}}+x)b)\beta 2}{a}\right))){\mathrm{e}}^{-\frac{\int (\mathit{c}\mathit{1}({\coth }^{\mathit{n}\mathit{1}}\left(\mathit{\text{\_b}}\lambda 1\right))+\mathit{c}\mathit{2}({\coth }^{\mathit{n}\mathit{2}}\left(\frac{(ay-(-\mathit{\text{\_b}}+x)b)\lambda 2}{a}\right)))d\mathit{\text{\_b}}}{a}}}{a}d\mathit{\text{\_b}}+\mathit{\text{\_F1}}\left(\frac{ay-bx}{a}\right)){\mathrm{e}}^{{\int }_{}^x\frac{\mathit{c}\mathit{1}({\coth }^{\mathit{n}\mathit{1}}\left(\mathit{\text{\_a}}\lambda 1\right))+\mathit{c}\mathit{2}({\coth }^{\mathit{n}\mathit{2}}\left(\frac{(ay-(-\mathit{\text{\_a}}+x)b)\lambda 2}{a}\right))}{a}d\mathit{\text{\_a}}}$$

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7.5.10.4 [1270] Problem 4

problem number 1270

Added April 3, 2019.

Problem Chapter 5.4.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{\coth }^n(\lambda x)w_x+b{\coth }^m(\mu x)w_y=c{\coth }^k(\nu x)w+p{\coth }^s(\beta y)$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to \exp ({\int }_1^x\frac{c{\coth }^{-n}(\lambda K[2])\coth (\nu K[2])}{a}dK[2])({\int }_1^x\frac{\exp (-{\int }_1^{K[3]}\frac{c{\coth }^{-n}(\lambda K[2])\coth (\nu K[2])}{a}dK[2])p{\coth }^{-n}(\lambda K[3]){\coth }^s\left(\beta (y-{\int }_1^x\frac{b{\coth }^{-n}(\lambda K[1]){\coth }^m(\mu K[1])}{a}dK[1]+{\int }_1^{K[3]}\frac{b{\coth }^{-n}(\lambda K[1]){\coth }^m(\mu K[1])}{a}dK[1])\right)}{a}dK[3]+c_1(y-{\int }_1^x\frac{b{\coth }^{-n}(\lambda K[1]){\coth }^m(\mu K[1])}{a}dK[1]))\}\right\}$$

Maple ✓

restart; 
pde :=  a*coth(lambda*x)^n*diff(w(x,y),x)+ b*coth(mu*x)^m*diff(w(x,y),y) = c*coth(nu*x)*w(x,y)+p*coth(beta*y)^s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=({\int }_{}^x\frac{p{\left(\frac{\cosh \left(\mathit{\text{\_f}}\lambda \right)}{\sinh \left(\mathit{\text{\_f}}\lambda \right)}\right)}^{-n}{\left(\frac{\cosh \left(\frac{(ay+b(\int {\left(\frac{\cosh \left(\mathit{\text{\_f}}\mu \right)}{\sinh \left(\mathit{\text{\_f}}\mu \right)}\right)}^m{\left(\frac{\cosh \left(\mathit{\text{\_f}}\lambda \right)}{\sinh \left(\mathit{\text{\_f}}\lambda \right)}\right)}^{-n}d\mathit{\text{\_f}})-b(\int {\left(\frac{\cosh \left(\lambda x\right)}{\sinh \left(\lambda x\right)}\right)}^{-n}{\left(\frac{\cosh \left(\mu x\right)}{\sinh \left(\mu x\right)}\right)}^{m}dx))\beta }{a}\right)}{\sinh \left(\frac{(ay+b(\int {\left(\frac{\cosh \left(\mathit{\text{\_f}}\mu \right)}{\sinh \left(\mathit{\text{\_f}}\mu \right)}\right)}^m{\left(\frac{\cosh \left(\mathit{\text{\_f}}\lambda \right)}{\sinh \left(\mathit{\text{\_f}}\lambda \right)}\right)}^{-n}d\mathit{\text{\_f}})-b(\int {\left(\frac{\cosh \left(\lambda x\right)}{\sinh \left(\lambda x\right)}\right)}^{-n}{\left(\frac{\cosh \left(\mu x\right)}{\sinh \left(\mu x\right)}\right)}^{m}dx))\beta }{a}\right)}\right)}^s{\mathrm{e}}^{-\frac{c(\int \frac{{\left(\frac{\cosh \left(\mathit{\text{\_f}}\lambda \right)}{\sinh \left(\mathit{\text{\_f}}\lambda \right)}\right)}^{-n}\cosh \left(\mathit{\text{\_f}}\nu \right)}{\sinh \left(\mathit{\text{\_f}}\nu \right)}d\mathit{\text{\_f}})}{a}}}{a}d\mathit{\text{\_f}}+\mathit{\text{\_F1}}\left(\frac{ay-b(\int {\left(\frac{\cosh \left(\lambda x\right)}{\sinh \left(\lambda x\right)}\right)}^{-n}{\left(\frac{\cosh \left(\mu x\right)}{\sinh \left(\mu x\right)}\right)}^{m}dx)}{a}\right)){\mathrm{e}}^{\int \frac{c({\coth }^{-n}\left(\lambda x\right))\coth \left(\nu x\right)}{a}dx}$$

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7.5.10.5 [1271] Problem 5

problem number 1271

Added April 3, 2019.

Problem Chapter 5.4.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{\coth }^n(\lambda x)w_x+b{\coth }^m(\mu x)w_y=c{\coth }^k(\nu y)w+p{\coth }^s(\beta x)$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to \exp ({\int }_1^x\frac{c{\coth }^{-n}(\lambda K[2])\coth \left(\nu (y-{\int }_1^x\frac{b{\coth }^{-n}(\lambda K[1]){\coth }^m(\mu K[1])}{a}dK[1]+{\int }_1^{K[2]}\frac{b{\coth }^{-n}(\lambda K[1]){\coth }^m(\mu K[1])}{a}dK[1])\right)}{a}dK[2])({\int }_1^x\frac{\exp (-{\int }_1^{K[3]}\frac{c{\coth }^{-n}(\lambda K[2])\coth \left(\nu (y-{\int }_1^x\frac{b{\coth }^{-n}(\lambda K[1]){\coth }^m(\mu K[1])}{a}dK[1]+{\int }_1^{K[2]}\frac{b{\coth }^{-n}(\lambda K[1]){\coth }^m(\mu K[1])}{a}dK[1])\right)}{a}dK[2])p{\coth }^s(\beta K[3]){\coth }^{-n}(\lambda K[3])}{a}dK[3]+c_1(y-{\int }_1^x\frac{b{\coth }^{-n}(\lambda K[1]){\coth }^m(\mu K[1])}{a}dK[1]))\}\right\}$$

Maple ✓

restart; 
pde :=  a*coth(lambda*x)^n*diff(w(x,y),x)+ b*coth(mu*x)^m*diff(w(x,y),y) = c*coth(nu*y)*w(x,y)+p*coth(beta*x)^s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=({\int }_{}^x\frac{p{\left(\frac{\cosh \left(\mathit{\text{\_f}}\beta \right)}{\sinh \left(\mathit{\text{\_f}}\beta \right)}\right)}^s{\left(\frac{\cosh \left(\mathit{\text{\_f}}\lambda \right)}{\sinh \left(\mathit{\text{\_f}}\lambda \right)}\right)}^{-n}{\mathrm{e}}^{-\frac{c(\int \frac{{\left(\frac{\cosh \left(\mathit{\text{\_f}}\lambda \right)}{\sinh \left(\mathit{\text{\_f}}\lambda \right)}\right)}^{-n}\cosh \left(\frac{(ay+b(\int {\left(\frac{\cosh \left(\mathit{\text{\_f}}\mu \right)}{\sinh \left(\mathit{\text{\_f}}\mu \right)}\right)}^m{\left(\frac{\cosh \left(\mathit{\text{\_f}}\lambda \right)}{\sinh \left(\mathit{\text{\_f}}\lambda \right)}\right)}^{-n}d\mathit{\text{\_f}})-b(\int {\left(\frac{\cosh \left(\lambda x\right)}{\sinh \left(\lambda x\right)}\right)}^{-n}{\left(\frac{\cosh \left(\mu x\right)}{\sinh \left(\mu x\right)}\right)}^{m}dx))\nu }{a}\right)}{\sinh \left(\frac{(ay+b(\int {\left(\frac{\cosh \left(\mathit{\text{\_f}}\mu \right)}{\sinh \left(\mathit{\text{\_f}}\mu \right)}\right)}^m{\left(\frac{\cosh \left(\mathit{\text{\_f}}\lambda \right)}{\sinh \left(\mathit{\text{\_f}}\lambda \right)}\right)}^{-n}d\mathit{\text{\_f}})-b(\int {\left(\frac{\cosh \left(\lambda x\right)}{\sinh \left(\lambda x\right)}\right)}^{-n}{\left(\frac{\cosh \left(\mu x\right)}{\sinh \left(\mu x\right)}\right)}^{m}dx))\nu }{a}\right)}d\mathit{\text{\_f}})}{a}}}{a}d\mathit{\text{\_f}}+\mathit{\text{\_F1}}\left(\frac{ay-b(\int {\left(\frac{\cosh \left(\lambda x\right)}{\sinh \left(\lambda x\right)}\right)}^{-n}{\left(\frac{\cosh \left(\mu x\right)}{\sinh \left(\mu x\right)}\right)}^{m}dx)}{a}\right)){\mathrm{e}}^{{\int }_{}^x\frac{c({\coth }^{-n}\left(\mathit{\text{\_b}}\lambda \right))\coth \left(\frac{(-b(\int {\left(\frac{\cosh \left(\lambda x\right)}{\sinh \left(\lambda x\right)}\right)}^{-n}{\left(\frac{\cosh \left(\mu x\right)}{\sinh \left(\mu x\right)}\right)}^{m}dx)+(y+\int \frac{b({\coth }^{-n}\left(\mathit{\text{\_b}}\lambda \right))({\coth }^m\left(\mathit{\text{\_b}}\mu \right))}{a}d\mathit{\text{\_b}})a)\nu }{a}\right)}{a}d\mathit{\text{\_b}}}$$

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