7.7.17 6.4

7.7.17.1 [1692] Problem 1

problem number 1692

Added June 26, 2019.

Problem Chapter 7.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+a{w}_y+b{w}_z=c{\cot }^k(\lambda x)+s$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to c_1(y-ax,z-cx)-\frac{c{\cot }^{k+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{k+1}{2},\frac{k+3}{2},-{\cot }^2(\lambda x))}{k\lambda +\lambda }+sx\}\right\}$$

Maple ✓

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

$$w(x,y,z)=sx+\int c({\cot }^k\left(\lambda x\right))dx+\mathit{\text{\_F1}}(-ax+y,-bx+z)$$

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7.7.17.2 [1693] Problem 2

problem number 1693

Added June 26, 2019.

Problem Chapter 7.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$a{w}_x+b{w}_y+c\cot (\gamma z)w_z=k\cot (\lambda x)+s\cot (\beta y)$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to \frac{a\lambda s\log (\tan (\beta y))+a\lambda s\log (\cos (\beta y))+b\beta k\log (\sin (\lambda x))}{ab\beta \lambda }+c_1(y-\frac{bx}{a},\frac{\log (\sec (\gamma z))}{\gamma }-\frac{cx}{a})\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\frac{2ab\beta \lambda \mathit{\text{\_F1}}(\frac{-ay+bx}{b},\frac{-2c\gamma y+b\ln ({\cot }^2\left(\gamma z\right)+1)-2b\ln (\cot \left(\gamma z\right))}{2c\gamma })-a\lambda s\ln ({\cot }^2\left(\beta y\right)+1)-b\beta k\ln ({\cot }^2\left(\lambda x\right)+1)}{2ab\beta \lambda }$$

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7.7.17.3 [1694] Problem 3

problem number 1694

Added June 26, 2019.

Problem Chapter 7.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+a{\cot }^n(\beta x)w_y+b{\cot }^k(\lambda x)w_z=c{\cot }^m(\gamma x)+s$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to c_1(\frac{b{\cot }^{k+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{k+1}{2},\frac{k+3}{2},-{\cot }^2(\lambda x))}{k\lambda +\lambda }+z,\frac{a{\cot }^{n+1}(\beta x)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{n+1}{2},\frac{n+3}{2},-{\cot }^2(\beta x))}{\beta n+\beta }+y)-\frac{c{\cot }^{m+1}(\gamma x)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{m+1}{2},\frac{m+3}{2},-{\cot }^2(\gamma x))}{\gamma m+\gamma }+sx\}\right\}$$

Maple ✓

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

$$w(x,y,z)=sx+\int c({\cot }^m\left(\gamma x\right))dx+\mathit{\text{\_F1}}(y-(\int a({\cot }^n\left(\beta x\right))dx),z-(\int b({\cot }^k\left(\lambda x\right))dx))$$

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7.7.17.4 [1695] Problem 4

problem number 1695

Added June 26, 2019.

Problem Chapter 7.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+a{\cot }^n(\lambda x)w_y+b{\cot }^m(\beta y)w_z=c{\cot }^k(\gamma y)+s{\cot }^r(\mu z)$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to {\int }_1^x(c{\cot }^k\left(\frac{\gamma (a\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{n+1}{2},\frac{n+3}{2},-{\cot }^2(\lambda x)){\cot }^{n+1}(\lambda x)+\lambda (n+1)y-a{\cot }^{n+1}(\lambda K[1])\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{n+1}{2},\frac{n+3}{2},-{\cot }^2(\lambda K[1])))}{\lambda (n+1)}\right)+s{\cot }^r\left(\frac{\mu (b\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{m+1}{2},\frac{m+3}{2},-{\cot }^2(\beta x)){\cot }^{m+1}(\beta x)+\beta (m+1)z-b{\cot }^{m+1}(\beta K[1])\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{m+1}{2},\frac{m+3}{2},-{\cot }^2(\beta K[1])))}{\beta (m+1)}\right))dK[1]+c_1(\frac{b{\cot }^{m+1}(\beta x)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{m+1}{2},\frac{m+3}{2},-{\cot }^2(\beta x))}{\beta m+\beta }+z,\frac{a{\cot }^{n+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{n+1}{2},\frac{n+3}{2},-{\cot }^2(\lambda x))}{\lambda n+\lambda }+y)\}\right\}$$

Maple ✓

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

$$w(x,y,z)={\int }_{}^x(c{\left(\frac{\cot \left((y-(\int a({\cot }^n\left(\lambda x\right))dx))\gamma \right)\cot \left(a\gamma (\int ({\cot }^n\left(\mathit{\text{\_f}}\lambda \right))d\mathit{\text{\_f}})\right)-1}{\cot \left((y-(\int a({\cot }^n\left(\lambda x\right))dx))\gamma \right)+\cot \left(a\gamma (\int ({\cot }^n\left(\mathit{\text{\_f}}\lambda \right))d\mathit{\text{\_f}})\right)}\right)}^k+s{\left(\frac{\cot \left((z-(\int b({\cot }^m\left(\beta x\right))dx))\mu \right)\cot \left(b\mu (\int ({\cot }^m\left(\mathit{\text{\_f}}\beta \right))d\mathit{\text{\_f}})\right)-1}{\cot \left((z-(\int b({\cot }^m\left(\beta x\right))dx))\mu \right)+\cot \left(b\mu (\int ({\cot }^m\left(\mathit{\text{\_f}}\beta \right))d\mathit{\text{\_f}})\right)}\right)}^r)d\mathit{\text{\_f}}+\mathit{\text{\_F1}}(y-(\int a({\cot }^n\left(\lambda x\right))dx),z-(\int b({\cot }^m\left(\beta x\right))dx))$$

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7.7.17.5 [1696] Problem 5

problem number 1696

Added June 26, 2019.

Problem Chapter 7.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$a{w}_x+b\cot (\beta x)w_y+c\cot (\lambda x)w_z=k\cot (\gamma z)$$

Mathematica ✓

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

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

Maple ✓

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

$$w(x,y,z)={\int }_{}^x\frac{(\cot \left(\frac{(2a\lambda z+c\ln ({\cot }^2\left(\lambda x\right)+1))\gamma }{2a\lambda }\right)\cot \left(\frac{c\gamma \ln ({\cot }^2\left(\mathit{\text{\_a}}\lambda \right)+1)}{2a\lambda }\right)+1)k}{(-\cot \left(\frac{(2a\lambda z+c\ln ({\cot }^2\left(\lambda x\right)+1))\gamma }{2a\lambda }\right)+\cot \left(\frac{c\gamma \ln ({\cot }^2\left(\mathit{\text{\_a}}\lambda \right)+1)}{2a\lambda }\right))a}d\mathit{\text{\_a}}+\mathit{\text{\_F1}}(\frac{2a\beta y+b\ln ({\cot }^2\left(\beta x\right)+1)}{2a\beta },\frac{2a\lambda z+c\ln ({\cot }^2\left(\lambda x\right)+1)}{2a\lambda })$$

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