6.8.20 7.2

6.8.20.1 [1875] Problem 1

problem number 1875

Added Nov 30, 2019.

Problem Chapter 8.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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*ArcCos[beta*x]^n * w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(y-ax,z-bx)\exp \left(\frac{c{\cos }^{-1}(\beta x{)}^n{({\cos }^{-1}(\beta x{)}^2)}^{-n}({(-i{\cos }^{-1}(\beta x))}^n\mathrm{Gamma}(n+1,i{\cos }^{-1}(\beta x))+{(i{\cos }^{-1}(\beta x))}^n\mathrm{Gamma}(n+1,-i{\cos }^{-1}(\beta x)))}{2\beta }\right)\}\right\}$$

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

$$w(x,y,z)=\mathit{\_}\mathit{F}\mathit{1}(-ax+y,-bx+z){\mathrm{e}}^{\int c\arccos {\left(\beta x\right)}^{n}dx}$$

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6.8.20.2 [1876] Problem 2

problem number 1876

Added Nov 30, 2019.

Problem Chapter 8.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a_1{w}_x+a_2{w}_y+a_3{w}_z=(b_1\arccos ({\lambda }_{1}x)+b_2\arccos ({\lambda }_{2}y)+b_3\arccos ({\lambda }_{3}z))w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a1*D[w[x,y,z],x]+a2*D[w[x,y,z],y]+a3*D[w[x,y,z],z]== (b1*ArcCos[lambda1*x]+b2*ArcCos[lambda2*y]+b3*ArcCos[lambda3*z] ) * w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(y-\frac{\text{a2}x}{\text{a1}},z-\frac{\text{a3}x}{\text{a1}})\exp (-\frac{\text{b1}\sqrt{1-{\text{lambda1}}^2{x}^2}}{\text{a1}\text{lambda1}}+\frac{\text{b1}x{\cos }^{-1}(\text{lambda1}x)}{\text{a1}}+\frac{\text{b2}x{\sin }^{-1}(\text{lambda2}y)}{\text{a1}}+\frac{\text{b2}x{\cos }^{-1}(\text{lambda2}y)}{\text{a1}}+\frac{\text{b3}x{\sin }^{-1}(\text{lambda3}z)}{\text{a1}}+\frac{\text{b3}x{\cos }^{-1}(\text{lambda3}z)}{\text{a1}}-\frac{\text{b2}\sqrt{1-{\text{lambda2}}^2{y}^2}}{\text{a2}\text{lambda2}}-\frac{\text{b2}y{\sin }^{-1}(\text{lambda2}y)}{\text{a2}}-\frac{\text{b3}\sqrt{1-{\text{lambda3}}^2{z}^2}}{\text{a3}\text{lambda3}}-\frac{\text{b3}z{\sin }^{-1}(\text{lambda3}z)}{\text{a3}})\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=  a__1*diff(w(x,y,z),x)+ a__2*diff(w(x,y,z),y)+ a__3*diff(w(x,y,z),z)= (b__1*arccos(lambda__1*x)+b__2*arccos(lambda__2*y)+b__3*arccos(lambda__3*z))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\_}\mathit{F}\mathit{1}(\frac{y{a}_1-x{a}_2}{a_1},\frac{z{a}_1-a_{3}x}{a_1}){\mathrm{e}}^{\frac{-\sqrt{-y^2{\lambda }_2^2+1}{a}_1{a}_3{b}_2{\lambda }_1{\lambda }_3+(-\sqrt{-z^2{\lambda }_3^2+1}{a}_1{a}_2{b}_3{\lambda }_1+(-\sqrt{-{\lambda }_1^2{x}^2+1}{a}_2{a}_3{b}_1+(a_2{a}_3{b}_{1}x\arccos \left(x{\lambda }_1\right)+(a_2{b}_{3}z\arccos \left(z{\lambda }_3\right)+a_3{b}_{2}y\arccos \left(y{\lambda }_2\right))a_1){\lambda }_1){\lambda }_3){\lambda }_2}{a_1{a}_2{a}_3{\lambda }_1{\lambda }_2{\lambda }_3}}$$

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6.8.20.3 [1877] Problem 3

problem number 1877

Added Nov 30, 2019.

Problem Chapter 8.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]==s*ArcCos[gamma*x]^m * w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
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)= s*arccos(gamma*x)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.8.20.4 [1878] Problem 4

problem number 1878

Added Nov 30, 2019.

Problem Chapter 8.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]==s* w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

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

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

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6.8.20.5 [1879] Problem 5

problem number 1879

Added Nov 30, 2019.

Problem Chapter 8.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]==s* ArcCos[gamma*x]^m*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
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)= s*arccos(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{\_}\mathit{F}\mathit{1}(\frac{-\frac{(-\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+\frac{3}{2}}+(n+2)\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\lambda x\right)))\sqrt{-{\lambda }^2{x}^2+1}b{2}^n{2}^{-n}}{\sqrt{\arccos \left(\lambda x\right)}}-(n+2)(-bx{2}^n{2}^{-n}\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{\_}\mathit{Z}b\lambda n\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)+2\mathit{\_}\mathit{Z}b\lambda \mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)-\mathit{\_}\mathit{b}a\lambda n\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}+a\lambda ny\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-a\lambda n(\int \frac{b\arccos {\left(\lambda x\right)}^n}{a}dx)\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-2\mathit{\_}\mathit{b}a\lambda \sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}+2a\lambda y\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-2a\lambda (\int \frac{b\arccos {\left(\lambda x\right)}^n}{a}dx)\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}bn\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))+\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}b\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)-\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}b\arccos {\left(\mathit{\_}\mathit{Z}\lambda \right)}^{n+\frac{3}{2}}-2\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}b\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)))\right)}^{-n}d\mathit{\_}\mathit{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}){\mathrm{e}}^{{\int }_{}^y\frac{s\arccos {\left(\gamma \mathrm{RootOf}(\mathit{\_}\mathit{Z}b\lambda n\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)+2\mathit{\_}\mathit{Z}b\lambda \mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)-\mathit{\_}\mathit{a}a\lambda n\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}+a\lambda ny\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-a\lambda n(\int \frac{b\arccos {\left(\lambda x\right)}^n}{a}dx)\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-2\mathit{\_}\mathit{a}a\lambda \sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}+2a\lambda y\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-2a\lambda (\int \frac{b\arccos {\left(\lambda x\right)}^n}{a}dx)\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}bn\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))+\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}b\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)-\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}b\arccos {\left(\mathit{\_}\mathit{Z}\lambda \right)}^{n+\frac{3}{2}}-2\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}b\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)))\right)}^m\arccos {\left(\lambda \mathrm{RootOf}(\mathit{\_}\mathit{Z}b\lambda n\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)+2\mathit{\_}\mathit{Z}b\lambda \mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)-\mathit{\_}\mathit{a}a\lambda n\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}+a\lambda ny\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-a\lambda n(\int \frac{b\arccos {\left(\lambda x\right)}^n}{a}dx)\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-2\mathit{\_}\mathit{a}a\lambda \sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}+2a\lambda y\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-2a\lambda (\int \frac{b\arccos {\left(\lambda x\right)}^n}{a}dx)\sqrt{\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)}-\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}bn\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))+\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}b\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right))\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)-\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}b\arccos {\left(\mathit{\_}\mathit{Z}\lambda \right)}^{n+\frac{3}{2}}-2\sqrt{-{\mathit{\_}\mathit{Z}}^2{\lambda }^2+1}b\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(\mathit{\_}\mathit{Z}\lambda \right)))\right)}^{-n}}{b}d\mathit{\_}\mathit{a}}$$

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6.8.20.6 [1880] Problem 6

problem number 1880

Added Nov 30, 2019.

Problem Chapter 8.7.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]==s* w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
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)= s*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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