7.6.20 7.2

7.6.20.1 [1537] Problem 1

problem number 1537

Added May 31, 2019.

Problem Chapter 6.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$a{w}_x+b{w}_y+c{\arccos }^n(\lambda x){\arccos }^k(\beta z)w_z=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*ArcCos[lambda*x]^n*ArcCos[beta*z]^k*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(y-\frac{bx}{a},-{\int }_1^x\frac{c{\cos }^{-1}(\lambda K[1]{)}^n}{a}dK[1]+\frac{{\cos }^{-1}(\beta z{)}^{-k}({(-i{\cos }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,-i{\cos }^{-1}(\beta z))+{(i{\cos }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,i{\cos }^{-1}(\beta z)))}{2\beta })\}\right\}$$

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*arccos(lambda*x)^n*arccos(beta*z)^k*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{ay-bx}{a},\frac{\sqrt{\pi }(-\frac{\sqrt{-{\lambda }^2{x}^2+1}{2}^{-n}\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(\lambda x\right))\sqrt{\arccos \left(\lambda x\right)}}{\sqrt{\pi }(n+2)}+\frac{\sqrt{-{\lambda }^2{x}^2+1}{2}^{-n}\arccos {\left(\lambda x\right)}^{n+1}}{\sqrt{\pi }(n+2)}-\frac{3(\frac{2n}{3}+\frac{4}{3})(\lambda x\arccos \left(\lambda x\right)-\sqrt{-{\lambda }^2{x}^2+1})2^{-n-1}\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\lambda x\right))}{\sqrt{\pi }(n+2)\sqrt{\arccos \left(\lambda x\right)}})2^n}{\lambda }+\frac{(2\beta kz{2}^{k-1}\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right))\arccos \left(\beta z\right)-4\beta z{2}^{k-1}\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right))\arccos \left(\beta z\right)-2\sqrt{-{\beta }^2{z}^2+1}k{2}^{k-1}\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right))-\sqrt{-{\beta }^2{z}^2+1}{2}^k\mathrm{LommelS}1(-k+\frac{3}{2},\frac{3}{2},\arccos \left(\beta z\right))\arccos \left(\beta z\right)+\sqrt{-{\beta }^2{z}^2+1}{2}^k\arccos {\left(\beta z\right)}^{-k+1}\sqrt{\arccos \left(\beta z\right)}+4\sqrt{-{\beta }^2{z}^2+1}{2}^{k-1}\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right)))a{2}^{-k}}{(k-2)\beta c\sqrt{\arccos \left(\beta z\right)}})$$

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7.6.20.2 [1538] Problem 2

problem number 1538

Added May 31, 2019.

Problem Chapter 6.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$a{w}_x+b{w}_y+c{\arccos }^n(\lambda x){\arccos }^m(\beta y){\arccos }^k(\gamma z)w_z=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*ArcCos[lambda*x]^n*ArcCos[beta*y]^m*ArcCos[gamma*z]^k*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(y-\frac{bx}{a},\frac{{\cos }^{-1}(\gamma z{)}^{-k}(-2\gamma {\cos }^{-1}(\gamma z{)}^k{\int }_1^x\frac{c{\cos }^{-1}(\lambda K[1]{)}^n\left(\left(\frac{a{\cos }^{-1}(\lambda K[1]{)}^{-n}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][{\int }_1^x\frac{c{\cos }^{-1}(\lambda K[1]{)}^n{\cos }^{-1}{\left(\beta (y+\frac{b(K[1]-x)}{a})\right)}^m}{a}dK[1],\{K[1],1,x\}]}{c}\right){}^{\frac{1}{m}}\right){}^m}{a}dK[1]+{(-i{\cos }^{-1}(\gamma z))}^k\mathrm{Gamma}(1-k,-i{\cos }^{-1}(\gamma z))+{(i{\cos }^{-1}(\gamma z))}^k\mathrm{Gamma}(1-k,i{\cos }^{-1}(\gamma z)))}{2\gamma })\}\right\}$$

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*arccos(lambda*x)^n*arccos(beta*y)^m*arccos(gamma1*z)^k*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{ay-bx}{a},\frac{-(k-2)c\gamma 1({\int }_{}^x\arccos {\left(\mathit{\text{\_a}}\lambda \right)}^n\arccos {\left(\frac{(ay-(-\mathit{\text{\_a}}+x)b)\beta }{a}\right)}^{m}d\mathit{\text{\_a}})+\frac{((k-2)\gamma 1z\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\gamma 1z\right))\arccos \left(\gamma 1z\right)+(-\mathrm{LommelS}1(-k+\frac{3}{2},\frac{3}{2},\arccos \left(\gamma 1z\right))\arccos \left(\gamma 1z\right)+\arccos {\left(\gamma 1z\right)}^{-k+\frac{3}{2}}+(-k+2)\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\gamma 1z\right)))\sqrt{-\gamma 1^2{z}^2+1})a{2}^k{2}^{-k}}{\sqrt{\arccos \left(\gamma 1z\right)}}}{(k-2)c\gamma 1})$$

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7.6.20.3 [1539] Problem 3

problem number 1539

Added May 31, 2019.

Problem Chapter 6.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$a{w}_x+b{\arccos }^n(\lambda x)w_y+c{\arccos }^k(\beta x)w_z=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*ArcCos[lambda*x]^n*D[w[x, y,z], y] +c*ArcCos[beta*x]^k*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(\frac{{({\cos }^{-1}(\beta x{)}^2)}^{-k}(-c{(i{\cos }^{-1}(\beta x))}^k{\cos }^{-1}(\beta x{)}^k\mathrm{Gamma}(k+1,-i{\cos }^{-1}(\beta x))-c{(-i{\cos }^{-1}(\beta x))}^k{\cos }^{-1}(\beta x{)}^k\mathrm{Gamma}(k+1,i{\cos }^{-1}(\beta x))+2a\beta z{({\cos }^{-1}(\beta x{)}^2)}^k)}{2a\beta },y-{\int }_1^x\frac{b{\cos }^{-1}(\lambda K[1]{)}^n}{a}dK[1])\}\right\}$$

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*arccos(lambda*x)^n*diff(w(x,y,z),y)+c*arccos(beta*x)^k*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(y+\frac{\sqrt{\pi }(-\frac{\sqrt{-{\lambda }^2{x}^2+1}{2}^{-n}\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(\lambda x\right))\sqrt{\arccos \left(\lambda x\right)}}{\sqrt{\pi }(n+2)}+\frac{\sqrt{-{\lambda }^2{x}^2+1}{2}^{-n}\arccos {\left(\lambda x\right)}^{n+1}}{\sqrt{\pi }(n+2)}-\frac{3(\frac{2n}{3}+\frac{4}{3})(\lambda x\arccos \left(\lambda x\right)-\sqrt{-{\lambda }^2{x}^2+1})2^{-n-1}\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\lambda x\right))}{\sqrt{\pi }(n+2)\sqrt{\arccos \left(\lambda x\right)}})b{2}^n}{a\lambda },z+\frac{\sqrt{\pi }(-\frac{\sqrt{-{\beta }^2{x}^2+1}{2}^{-k}\mathrm{LommelS}1(k+\frac{3}{2},\frac{3}{2},\arccos \left(\beta x\right))\sqrt{\arccos \left(\beta x\right)}}{\sqrt{\pi }(k+2)}+\frac{\sqrt{-{\beta }^2{x}^2+1}{2}^{-k}\arccos {\left(\beta x\right)}^{k+1}}{\sqrt{\pi }(k+2)}-\frac{3(\frac{2k}{3}+\frac{4}{3})(\beta x\arccos \left(\beta x\right)-\sqrt{-{\beta }^2{x}^2+1})2^{-k-1}\mathrm{LommelS}1(k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta x\right))}{\sqrt{\pi }(k+2)\sqrt{\arccos \left(\beta x\right)}})c{2}^k}{a\beta })$$

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7.6.20.4 [1540] Problem 4

problem number 1540

Added May 31, 2019.

Problem Chapter 6.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$a{w}_x+b{\arccos }^n(\lambda x)w_y+c{\arccos }^k(\beta z)w_z=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*ArcCos[lambda*x]^n*D[w[x, y,z], y] +c*ArcCos[beta*z]^k*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(-\frac{cx}{a}+\frac{{\cos }^{-1}(\beta z{)}^{-k}({(-i{\cos }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,-i{\cos }^{-1}(\beta z))+{(i{\cos }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,i{\cos }^{-1}(\beta z)))}{2\beta },y-{\int }_1^x\frac{b{\cos }^{-1}(\lambda K[1]{)}^n}{a}dK[1])\}\right\}$$

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*arccos(lambda*x)^n*diff(w(x,y,z),y)+c*arccos(beta*z)^k*diff(w(x,y,z),z)= 0; 
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{\frac{(-2(n+2)2^{-n-1}\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\lambda x\right))+(\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(\lambda x\right))\arccos \left(\lambda x\right)-\arccos {\left(\lambda x\right)}^{n+1}\sqrt{\arccos \left(\lambda x\right)})2^{-n})\sqrt{-{\lambda }^2{x}^2+1}b{2}^n}{\sqrt{\arccos \left(\lambda x\right)}}-(n+2)(-2bx{2}^n{2}^{-n-1}\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\lambda x\right))\sqrt{\arccos \left(\lambda x\right)}+ay)\lambda }{(n+2)a\lambda },\frac{-(k-2)\beta c({\int }_{}^y\arccos {\left(\lambda \mathrm{RootOf}(\mathit{\text{\_Z}}b\lambda n\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\text{\_Z}}\lambda \right))\arccos \left(\mathit{\text{\_Z}}\lambda \right)+2\mathit{\text{\_Z}}b\lambda \mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\text{\_Z}}\lambda \right))\arccos \left(\mathit{\text{\_Z}}\lambda \right)-\mathit{\text{\_b}}a\lambda n\sqrt{\arccos \left(\mathit{\text{\_Z}}\lambda \right)}+a\lambda ny\sqrt{\arccos \left(\mathit{\text{\_Z}}\lambda \right)}-a\lambda n(\int \frac{b\arccos {\left(\lambda x\right)}^n}{a}dx)\sqrt{\arccos \left(\mathit{\text{\_Z}}\lambda \right)}-2\mathit{\text{\_b}}a\lambda \sqrt{\arccos \left(\mathit{\text{\_Z}}\lambda \right)}+2a\lambda y\sqrt{\arccos \left(\mathit{\text{\_Z}}\lambda \right)}-2a\lambda (\int \frac{b\arccos {\left(\lambda x\right)}^n}{a}dx)\sqrt{\arccos \left(\mathit{\text{\_Z}}\lambda \right)}-\sqrt{-{\mathit{\text{\_Z}}}^2{\lambda }^2+1}bn\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\text{\_Z}}\lambda \right))+\sqrt{-{\mathit{\text{\_Z}}}^2{\lambda }^2+1}b\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(\mathit{\text{\_Z}}\lambda \right))\arccos \left(\mathit{\text{\_Z}}\lambda \right)-\sqrt{-{\mathit{\text{\_Z}}}^2{\lambda }^2+1}b\arccos {\left(\mathit{\text{\_Z}}\lambda \right)}^{n+\frac{3}{2}}-2\sqrt{-{\mathit{\text{\_Z}}}^2{\lambda }^2+1}b\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\text{\_Z}}\lambda \right)))\right)}^{-n}d\mathit{\text{\_b}})+\frac{((k-2)\beta z\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right))\arccos \left(\beta z\right)+(-\mathrm{LommelS}1(-k+\frac{3}{2},\frac{3}{2},\arccos \left(\beta z\right))\arccos \left(\beta z\right)+\arccos {\left(\beta z\right)}^{-k+\frac{3}{2}}+(-k+2)\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right)))\sqrt{-{\beta }^2{z}^2+1})b{2}^k{2}^{-k}}{\sqrt{\arccos \left(\beta z\right)}}}{(k-2)\beta c})$$

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7.6.20.5 [1541] Problem 5

problem number 1541

Added May 31, 2019.

Problem Chapter 6.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$a{w}_x+b{\arccos }^n(\lambda y)w_y+c{\arccos }^k(\beta z)w_z=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*ArcCos[lambda*y]^n*D[w[x, y,z], y] +c*ArcCos[beta*z]^k*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(-\frac{cx}{a}+\frac{{\cos }^{-1}(\beta z{)}^{-k}({(-i{\cos }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,-i{\cos }^{-1}(\beta z))+{(i{\cos }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,i{\cos }^{-1}(\beta z)))}{2\beta },{\int }_1^y{\cos }^{-1}(\lambda K[1]{)}^{-n}dK[1]-\frac{bx}{a})\}\right\}$$

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*arccos(lambda*y)^n*diff(w(x,y,z),y)+c*arccos(beta*z)^k*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(x+\frac{\sqrt{\pi }(\frac{\sqrt{-{\lambda }^2{y}^2+1}{2}^n\mathrm{LommelS}1(-n+\frac{3}{2},\frac{3}{2},\arccos \left(\lambda y\right))\sqrt{\arccos \left(\lambda y\right)}}{\sqrt{\pi }(n-2)}-\frac{\sqrt{-{\lambda }^2{y}^2+1}{2}^n\arccos {\left(\lambda y\right)}^{-n+1}}{\sqrt{\pi }(n-2)}+\frac{3(-\frac{2n}{3}+\frac{4}{3})(\lambda y\arccos \left(\lambda y\right)-\sqrt{-{\lambda }^2{y}^2+1})2^{n-1}\mathrm{LommelS}1(-n+\frac{1}{2},\frac{1}{2},\arccos \left(\lambda y\right))}{\sqrt{\pi }(n-2)\sqrt{\arccos \left(\lambda y\right)}})a{2}^{-n}}{b\lambda },\frac{\sqrt{\pi }(\frac{\sqrt{-{\lambda }^2{y}^2+1}{2}^n\mathrm{LommelS}1(-n+\frac{3}{2},\frac{3}{2},\arccos \left(\lambda y\right))\sqrt{\arccos \left(\lambda y\right)}}{\sqrt{\pi }(n-2)}-\frac{\sqrt{-{\lambda }^2{y}^2+1}{2}^n\arccos {\left(\lambda y\right)}^{-n+1}}{\sqrt{\pi }(n-2)}+\frac{3(-\frac{2n}{3}+\frac{4}{3})(\lambda y\arccos \left(\lambda y\right)-\sqrt{-{\lambda }^2{y}^2+1})2^{n-1}\mathrm{LommelS}1(-n+\frac{1}{2},\frac{1}{2},\arccos \left(\lambda y\right))}{\sqrt{\pi }(n-2)\sqrt{\arccos \left(\lambda y\right)}})2^{-n}}{\lambda }+\frac{(2\beta kz{2}^{k-1}\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right))\arccos \left(\beta z\right)-4\beta z{2}^{k-1}\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right))\arccos \left(\beta z\right)-2\sqrt{-{\beta }^2{z}^2+1}k{2}^{k-1}\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right))-\sqrt{-{\beta }^2{z}^2+1}{2}^k\mathrm{LommelS}1(-k+\frac{3}{2},\frac{3}{2},\arccos \left(\beta z\right))\arccos \left(\beta z\right)+\sqrt{-{\beta }^2{z}^2+1}{2}^k\arccos {\left(\beta z\right)}^{-k+1}\sqrt{\arccos \left(\beta z\right)}+4\sqrt{-{\beta }^2{z}^2+1}{2}^{k-1}\mathrm{LommelS}1(-k+\frac{1}{2},\frac{1}{2},\arccos \left(\beta z\right)))b{2}^{-k}}{(k-2)\beta c\sqrt{\arccos \left(\beta z\right)}})$$

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