7.8.15 6.2

7.8.15.1 [1850] Problem 1

problem number 1850

Added Oct 18, 2019.

Problem Chapter 8.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+a{w}_y+b{w}_z=c{\cos }^n(\beta x)w$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to c_1(y-ax,z-bx)\exp (-\frac{c\sqrt{{\sin }^2(\beta x)}\csc (\beta x){\cos }^{n+1}(\beta x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{n+1}{2},\frac{n+3}{2},{\cos }^2(\beta x))}{\beta n+\beta })\}\right\}$$

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*cos(beta*x)^n*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(-ax+y,-bx+z){\mathrm{e}}^{\int c({\cos }^n\left(\beta x\right))dx}$$

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7.8.15.2 [1851] Problem 2

problem number 1851

Added Oct 18, 2019.

Problem Chapter 8.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to e^{\frac{k\sin (\lambda x)}{a\lambda }+\frac{s\sin (\gamma y)}{b\gamma }}{c}_1(y-\frac{bx}{a},-\frac{{\cosh }^{-1}(-\frac{\sec (\beta z)(2(2\sec (\beta z)\sqrt{{\sin }^2(\beta z){\cos }^2(\beta z){\sinh }^2\left(\frac{\beta cx}{a}\right)(\cosh \left(\frac{4\beta cx}{a}\right)-\sinh \left(\frac{4\beta cx}{a}\right))}+{\sinh }^3\left(\frac{\beta cx}{a}\right)+\sinh \left(\frac{\beta cx}{a}\right))-2{\cosh }^3\left(\frac{\beta cx}{a}\right)+(1-3\cosh \left(\frac{2\beta cx}{a}\right))\cosh \left(\frac{\beta cx}{a}\right)+6\sinh \left(\frac{\beta cx}{a}\right){\cosh }^2\left(\frac{\beta cx}{a}\right))}{4\cosh \left(\frac{2\beta cx}{a}\right)-4\sinh \left(\frac{2\beta cx}{a}\right)})}{\beta })\}\right\}$$

Maple ✓

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

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

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7.8.15.3 [1852] Problem 3

problem number 1852

Added Oct 18, 2019.

Problem Chapter 8.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+a{\cos }^n(\beta x)w_y+b{\cos }^k(\lambda x)w_z=c{\cos }^m(\gamma x)w$$

Mathematica ✓

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

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

Maple ✓

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

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

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7.8.15.4 [1853] Problem 4

problem number 1853

Added Oct 18, 2019.

Problem Chapter 8.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$w_x+a{\cos }^n(\beta x)w_y+b{\cos }^m(\gamma y)w_z=(c{\cos }^k(\gamma y)+s{\cos }^r(\mu z))w$$

Mathematica ✗

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

Failed

Maple ✓

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

$$w(x,y,z)=\mathit{\text{\_F1}}(y-(\int a({\cos }^n\left(\beta x\right))dx),z-\left({\int }_{}^{x}b({\cos }^m\left((a(\int ({\cos }^n\left(\mathit{\text{\_b}}\beta \right))d\mathit{\text{\_b}})+y-(\int a({\cos }^n\left(\beta x\right))dx))\gamma \right))d\mathit{\text{\_b}}\right)){\mathrm{e}}^{{\int }_{}^x(c({\cos }^k\left((-y-(\int a({\cos }^n\left(\mathit{\text{\_g}}\beta \right))d\mathit{\text{\_g}})+\int a({\cos }^n\left(\beta x\right))dx)\gamma \right))+s({\cos }^r\left((z+\int b({\cos }^m\left((a(\int ({\cos }^n\left(\mathit{\text{\_g}}\beta \right))d\mathit{\text{\_g}})+y-(\int a({\cos }^n\left(\beta x\right))dx))\gamma \right))d\mathit{\text{\_g}}-\left({\int }^{x}b({\cos }^m\left((a(\int ({\cos }^n\left(\mathit{\text{\_b}}\beta \right))d\mathit{\text{\_b}})+y-(\int a({\cos }^n\left(\beta x\right))dx))\gamma \right))d\mathit{\text{\_b}}\right))\mu \right)))d\mathit{\text{\_g}}}$$

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7.8.15.5 [1854] Problem 5

problem number 1854

Added Oct 18, 2019.

Problem Chapter 8.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a{w}_x+b\cos (\beta y)w_y+c\cos (\lambda x)w_z=k\cos (\gamma z)w$$

Mathematica ✓

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

$$\begin{array}{rl}& \{w(x,y,z)\to c_1(-\frac{{\cosh }^{-1}(-\frac{\sec (\beta y)(2(2\sec (\beta y)\sqrt{{\sin }^2(\beta y){\cos }^2(\beta y){\sinh }^2\left(\frac{b\beta x}{a}\right)(\cosh \left(\frac{4b\beta x}{a}\right)-\sinh \left(\frac{4b\beta x}{a}\right))}+{\sinh }^3\left(\frac{b\beta x}{a}\right)+\sinh \left(\frac{b\beta x}{a}\right))-2{\cosh }^3\left(\frac{b\beta x}{a}\right)+(1-3\cosh \left(\frac{2b\beta x}{a}\right))\cosh \left(\frac{b\beta x}{a}\right)+6\sinh \left(\frac{b\beta x}{a}\right){\cosh }^2\left(\frac{b\beta x}{a}\right))}{4\cosh \left(\frac{2b\beta x}{a}\right)-4\sinh \left(\frac{2b\beta x}{a}\right)})}{\beta },\frac{c\sqrt{{\sin }^2(\lambda x)}\csc (\lambda x){\cos }^{m+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))}{a\lambda m+a\lambda }+z)\exp ({\int }_1^x\frac{k\cos \left(\frac{\gamma (c\csc (\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))\sqrt{{\sin }^2(\lambda x)}{\cos }^{m+1}(\lambda x)+a\lambda (m+1)z-c{\cos }^{m+1}(\lambda K[1])\csc (\lambda K[1])\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda K[1]))\sqrt{{\sin }^2(\lambda K[1])})}{a\lambda (m+1)}\right)}{a}dK[1])\}\\ & \{w(x,y,z)\to c_1(-\frac{{\cosh }^{-1}(-\frac{\sec (\beta y)(2(2\sec (\beta y)\sqrt{{\sin }^2(\beta y){\cos }^2(\beta y){\sinh }^2\left(\frac{b\beta x}{a}\right)(\cosh \left(\frac{4b\beta x}{a}\right)-\sinh \left(\frac{4b\beta x}{a}\right))}+{\sinh }^3\left(\frac{b\beta x}{a}\right)+\sinh \left(\frac{b\beta x}{a}\right))-2{\cosh }^3\left(\frac{b\beta x}{a}\right)+(1-3\cosh \left(\frac{2b\beta x}{a}\right))\cosh \left(\frac{b\beta x}{a}\right)+6\sinh \left(\frac{b\beta x}{a}\right){\cosh }^2\left(\frac{b\beta x}{a}\right))}{4\cosh \left(\frac{2b\beta x}{a}\right)-4\sinh \left(\frac{2b\beta x}{a}\right)})}{\beta },\frac{c\sqrt{{\sin }^2(\lambda x)}\csc (\lambda x){\cos }^{m+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))}{a\lambda m+a\lambda }+z)\exp ({\int }_1^x\frac{k\cos \left(\frac{\gamma (c\csc (\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))\sqrt{{\sin }^2(\lambda x)}{\cos }^{m+1}(\lambda x)+a\lambda (m+1)z-c{\cos }^{m+1}(\lambda K[2])\csc (\lambda K[2])\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda K[2]))\sqrt{{\sin }^2(\lambda K[2])})}{a\lambda (m+1)}\right)}{a}dK[2])\}\\ & \{w(x,y,z)\to c_1(-\frac{{\cosh }^{-1}(-\frac{\sec (\beta y)(2(2\sec (\beta y)\sqrt{{\sin }^2(\beta y){\cos }^2(\beta y){\sinh }^2\left(\frac{b\beta x}{a}\right)(\cosh \left(\frac{4b\beta x}{a}\right)-\sinh \left(\frac{4b\beta x}{a}\right))}+{\sinh }^3\left(\frac{b\beta x}{a}\right)+\sinh \left(\frac{b\beta x}{a}\right))-2{\cosh }^3\left(\frac{b\beta x}{a}\right)+(1-3\cosh \left(\frac{2b\beta x}{a}\right))\cosh \left(\frac{b\beta x}{a}\right)+6\sinh \left(\frac{b\beta x}{a}\right){\cosh }^2\left(\frac{b\beta x}{a}\right))}{4\cosh \left(\frac{2b\beta x}{a}\right)-4\sinh \left(\frac{2b\beta x}{a}\right)})}{\beta },\frac{c\sqrt{{\sin }^2(\lambda x)}\csc (\lambda x){\cos }^{m+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))}{a\lambda m+a\lambda }+z)\exp ({\int }_1^x\frac{k\cos \left(\frac{\gamma (c\csc (\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))\sqrt{{\sin }^2(\lambda x)}{\cos }^{m+1}(\lambda x)+a\lambda (m+1)z-c{\cos }^{m+1}(\lambda K[3])\csc (\lambda K[3])\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda K[3]))\sqrt{{\sin }^2(\lambda K[3])})}{a\lambda (m+1)}\right)}{a}dK[3])\}\\ & \{w(x,y,z)\to c_1(-\frac{{\cosh }^{-1}(-\frac{\sec (\beta y)(2(2\sec (\beta y)\sqrt{{\sin }^2(\beta y){\cos }^2(\beta y){\sinh }^2\left(\frac{b\beta x}{a}\right)(\cosh \left(\frac{4b\beta x}{a}\right)-\sinh \left(\frac{4b\beta x}{a}\right))}+{\sinh }^3\left(\frac{b\beta x}{a}\right)+\sinh \left(\frac{b\beta x}{a}\right))-2{\cosh }^3\left(\frac{b\beta x}{a}\right)+(1-3\cosh \left(\frac{2b\beta x}{a}\right))\cosh \left(\frac{b\beta x}{a}\right)+6\sinh \left(\frac{b\beta x}{a}\right){\cosh }^2\left(\frac{b\beta x}{a}\right))}{4\cosh \left(\frac{2b\beta x}{a}\right)-4\sinh \left(\frac{2b\beta x}{a}\right)})}{\beta },\frac{c\sqrt{{\sin }^2(\lambda x)}\csc (\lambda x){\cos }^{m+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))}{a\lambda m+a\lambda }+z)\exp ({\int }_1^x\frac{k\cos \left(\frac{\gamma (c\csc (\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))\sqrt{{\sin }^2(\lambda x)}{\cos }^{m+1}(\lambda x)+a\lambda (m+1)z-c{\cos }^{m+1}(\lambda K[4])\csc (\lambda K[4])\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda K[4]))\sqrt{{\sin }^2(\lambda K[4])})}{a\lambda (m+1)}\right)}{a}dK[4])\}\\ & \{w(x,y,z)\to c_1(-\frac{{\cosh }^{-1}(-\frac{\sec (\beta y)(2(2\sec (\beta y)\sqrt{{\sin }^2(\beta y){\cos }^2(\beta y){\sinh }^2\left(\frac{b\beta x}{a}\right)(\cosh \left(\frac{4b\beta x}{a}\right)-\sinh \left(\frac{4b\beta x}{a}\right))}+{\sinh }^3\left(\frac{b\beta x}{a}\right)+\sinh \left(\frac{b\beta x}{a}\right))-2{\cosh }^3\left(\frac{b\beta x}{a}\right)+(1-3\cosh \left(\frac{2b\beta x}{a}\right))\cosh \left(\frac{b\beta x}{a}\right)+6\sinh \left(\frac{b\beta x}{a}\right){\cosh }^2\left(\frac{b\beta x}{a}\right))}{4\cosh \left(\frac{2b\beta x}{a}\right)-4\sinh \left(\frac{2b\beta x}{a}\right)})}{\beta },\frac{c\sqrt{{\sin }^2(\lambda x)}\csc (\lambda x){\cos }^{m+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))}{a\lambda m+a\lambda }+z)\exp ({\int }_1^x\frac{k\cos \left(\frac{\gamma (c\csc (\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))\sqrt{{\sin }^2(\lambda x)}{\cos }^{m+1}(\lambda x)+a\lambda (m+1)z-c{\cos }^{m+1}(\lambda K[5])\csc (\lambda K[5])\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda K[5]))\sqrt{{\sin }^2(\lambda K[5])})}{a\lambda (m+1)}\right)}{a}dK[5])\}\\ & \{w(x,y,z)\to c_1(-\frac{{\cosh }^{-1}(-\frac{\sec (\beta y)(2(2\sec (\beta y)\sqrt{{\sin }^2(\beta y){\cos }^2(\beta y){\sinh }^2\left(\frac{b\beta x}{a}\right)(\cosh \left(\frac{4b\beta x}{a}\right)-\sinh \left(\frac{4b\beta x}{a}\right))}+{\sinh }^3\left(\frac{b\beta x}{a}\right)+\sinh \left(\frac{b\beta x}{a}\right))-2{\cosh }^3\left(\frac{b\beta x}{a}\right)+(1-3\cosh \left(\frac{2b\beta x}{a}\right))\cosh \left(\frac{b\beta x}{a}\right)+6\sinh \left(\frac{b\beta x}{a}\right){\cosh }^2\left(\frac{b\beta x}{a}\right))}{4\cosh \left(\frac{2b\beta x}{a}\right)-4\sinh \left(\frac{2b\beta x}{a}\right)})}{\beta },\frac{c\sqrt{{\sin }^2(\lambda x)}\csc (\lambda x){\cos }^{m+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))}{a\lambda m+a\lambda }+z)\exp ({\int }_1^x\frac{k\cos \left(\frac{\gamma (c\csc (\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))\sqrt{{\sin }^2(\lambda x)}{\cos }^{m+1}(\lambda x)+a\lambda (m+1)z-c{\cos }^{m+1}(\lambda K[6])\csc (\lambda K[6])\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda K[6]))\sqrt{{\sin }^2(\lambda K[6])})}{a\lambda (m+1)}\right)}{a}dK[6])\}\\ & \{w(x,y,z)\to c_1(-\frac{{\cosh }^{-1}(-\frac{\sec (\beta y)(2(2\sec (\beta y)\sqrt{{\sin }^2(\beta y){\cos }^2(\beta y){\sinh }^2\left(\frac{b\beta x}{a}\right)(\cosh \left(\frac{4b\beta x}{a}\right)-\sinh \left(\frac{4b\beta x}{a}\right))}+{\sinh }^3\left(\frac{b\beta x}{a}\right)+\sinh \left(\frac{b\beta x}{a}\right))-2{\cosh }^3\left(\frac{b\beta x}{a}\right)+(1-3\cosh \left(\frac{2b\beta x}{a}\right))\cosh \left(\frac{b\beta x}{a}\right)+6\sinh \left(\frac{b\beta x}{a}\right){\cosh }^2\left(\frac{b\beta x}{a}\right))}{4\cosh \left(\frac{2b\beta x}{a}\right)-4\sinh \left(\frac{2b\beta x}{a}\right)})}{\beta },\frac{c\sqrt{{\sin }^2(\lambda x)}\csc (\lambda x){\cos }^{m+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))}{a\lambda m+a\lambda }+z)\exp ({\int }_1^x\frac{k\cos \left(\frac{\gamma (c\csc (\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))\sqrt{{\sin }^2(\lambda x)}{\cos }^{m+1}(\lambda x)+a\lambda (m+1)z-c{\cos }^{m+1}(\lambda K[7])\csc (\lambda K[7])\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda K[7]))\sqrt{{\sin }^2(\lambda K[7])})}{a\lambda (m+1)}\right)}{a}dK[7])\}\\ & \{w(x,y,z)\to c_1(-\frac{{\cosh }^{-1}(-\frac{\sec (\beta y)(2(2\sec (\beta y)\sqrt{{\sin }^2(\beta y){\cos }^2(\beta y){\sinh }^2\left(\frac{b\beta x}{a}\right)(\cosh \left(\frac{4b\beta x}{a}\right)-\sinh \left(\frac{4b\beta x}{a}\right))}+{\sinh }^3\left(\frac{b\beta x}{a}\right)+\sinh \left(\frac{b\beta x}{a}\right))-2{\cosh }^3\left(\frac{b\beta x}{a}\right)+(1-3\cosh \left(\frac{2b\beta x}{a}\right))\cosh \left(\frac{b\beta x}{a}\right)+6\sinh \left(\frac{b\beta x}{a}\right){\cosh }^2\left(\frac{b\beta x}{a}\right))}{4\cosh \left(\frac{2b\beta x}{a}\right)-4\sinh \left(\frac{2b\beta x}{a}\right)})}{\beta },\frac{c\sqrt{{\sin }^2(\lambda x)}\csc (\lambda x){\cos }^{m+1}(\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))}{a\lambda m+a\lambda }+z)\exp ({\int }_1^x\frac{k\cos \left(\frac{\gamma (c\csc (\lambda x)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda x))\sqrt{{\sin }^2(\lambda x)}{\cos }^{m+1}(\lambda x)+a\lambda (m+1)z-c{\cos }^{m+1}(\lambda K[8])\csc (\lambda K[8])\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},{\cos }^2(\lambda K[8]))\sqrt{{\sin }^2(\lambda K[8])})}{a\lambda (m+1)}\right)}{a}dK[8])\}\end{array}$$

Maple ✓

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

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{a\ln \left(\mathrm{RootOf}(\beta y-\arctan (\frac{{\mathit{\text{\_Z}}}^2{\mathrm{e}}^{\frac{2b\beta x}{a}}-1}{{\mathit{\text{\_Z}}}^2{\mathrm{e}}^{\frac{2b\beta x}{a}}+1},\frac{2\mathit{\text{\_Z}}{\mathrm{e}}^{\frac{b\beta x}{a}}}{{\mathit{\text{\_Z}}}^2{\mathrm{e}}^{\frac{2b\beta x}{a}}+1}))\right)}{b\beta },z-(\int \frac{c({\cos }^m\left(\lambda x\right))}{a}dx)){\mathrm{e}}^{{\int }_{}^x\frac{k\cos \left((-z-(\int \frac{c({\cos }^m\left(\mathit{\text{\_b}}\lambda \right))}{a}d\mathit{\text{\_b}})+\int \frac{c({\cos }^m\left(\lambda x\right))}{a}dx)\gamma \right)}{a}d\mathit{\text{\_b}}}$$

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7.8.15.6 [1855] Problem 6

problem number 1855

Added Oct 18, 2019.

Problem Chapter 8.6.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a_1{\cos }^{n_1}({\lambda }_{1}x)w_x+b_1{\cos }^{m_1}({\beta }_{1}y)w_y+c_1{\cos }^{k_1}({\gamma }_{1}z)w_z=(a_2{\cos }^{n_2}({\lambda }_{2}x)+b_2{\cos }^{m_2}({\beta }_{2}y)+c_2{\cos }^{k_2}({\gamma }_{2}z))w$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a1*Cos[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cos[beta1*y]^m1*D[w[x, y,z], y] + c1*Cos[gamma1*z]^k1*D[w[x,y,z],z]== (a2*Cos[lambda2*z]^n2 + b2*Cos[beta2*y]^m2 + c2*Cos[gamma2*z]^k2)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$Aborted

Maple ✓

restart; 
local gamma; 
pde :=  a1*cos(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*cos(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cos(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*cos(lambda2*z)^n2 + b2*cos(beta2*y)^m2 + c2*cos(gamma2*z)^k2)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(-(\int ({\cos }^{-\mathit{m}\mathit{1}}\left(\beta 1y\right))dy)+\int \frac{\mathit{b}\mathit{1}({\cos }^{-\mathit{k}\mathit{1}}\left(\gamma 1z\right))}{\mathit{c}\mathit{1}}dz,x-\left({\int }_{}^y\frac{\mathit{a}\mathit{1}({\cos }^{-\mathit{m}\mathit{1}}\left(\mathit{\text{\_f}}\beta 1\right))({\cos }^{\mathit{n}\mathit{1}}\left(\lambda 1\mathrm{RootOf}(\int ({\cos }^{-\mathit{m}\mathit{1}}\left(\mathit{\text{\_f}}\beta 1\right))d\mathit{\text{\_f}}-(\int ({\cos }^{-\mathit{m}\mathit{1}}\left(\beta 1y\right))dy)+\int \frac{\mathit{b}\mathit{1}({\cos }^{-\mathit{k}\mathit{1}}\left(\gamma 1z\right))}{\mathit{c}\mathit{1}}dz-\left({\int }^{\mathit{\text{\_Z}}}\frac{\mathit{b}\mathit{1}({\cos }^{-\mathit{k}\mathit{1}}\left(\mathit{\text{\_b}}\gamma 1\right))}{\mathit{c}\mathit{1}}d\mathit{\text{\_b}}\right))\right))}{\mathit{b}\mathit{1}}d\mathit{\text{\_f}}\right)){\mathrm{e}}^{{\int }_{}^y\frac{(\mathit{a}\mathit{2}({\cos }^{\mathit{n}\mathit{2}}\left(\lambda 2\mathrm{RootOf}(\int ({\cos }^{-\mathit{m}\mathit{1}}\left(\mathit{\text{\_f}}\beta 1\right))d\mathit{\text{\_f}}-(\int ({\cos }^{-\mathit{m}\mathit{1}}\left(\beta 1y\right))dy)+\int \frac{\mathit{b}\mathit{1}({\cos }^{-\mathit{k}\mathit{1}}\left(\gamma 1z\right))}{\mathit{c}\mathit{1}}dz-\left({\int }^{\mathit{\text{\_Z}}}\frac{\mathit{b}\mathit{1}({\cos }^{-\mathit{k}\mathit{1}}\left(\mathit{\text{\_a}}\gamma 1\right))}{\mathit{c}\mathit{1}}d\mathit{\text{\_a}}\right))\right))+\mathit{b}\mathit{2}({\cos }^{\mathit{m}\mathit{2}}\left(\mathit{\text{\_f}}\beta 2\right))+\mathit{c}\mathit{2}({\cos }^{\mathit{k}\mathit{2}}\left(\gamma 2\mathrm{RootOf}(\int ({\cos }^{-\mathit{m}\mathit{1}}\left(\mathit{\text{\_f}}\beta 1\right))d\mathit{\text{\_f}}-(\int ({\cos }^{-\mathit{m}\mathit{1}}\left(\beta 1y\right))dy)+\int \frac{\mathit{b}\mathit{1}({\cos }^{-\mathit{k}\mathit{1}}\left(\gamma 1z\right))}{\mathit{c}\mathit{1}}dz-\left({\int }^{\mathit{\text{\_Z}}}\frac{\mathit{b}\mathit{1}({\cos }^{-\mathit{k}\mathit{1}}\left(\mathit{\text{\_a}}\gamma 1\right))}{\mathit{c}\mathit{1}}d\mathit{\text{\_a}}\right))\right)))({\cos }^{-\mathit{m}\mathit{1}}\left(\mathit{\text{\_f}}\beta 1\right))}{\mathit{b}\mathit{1}}d\mathit{\text{\_f}}}$$

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