7.7.20 7.1

7.7.20.1 [1703] Problem 1

problem number 1703

Added June 26, 2019.

Problem Chapter 7.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

$$\left\{\{w(x,y,z)\to c_1(y-ax,z-bx)+\frac{{({\sin }^{-1}(\lambda x{)}^2)}^{-k}(-ic{(i{\sin }^{-1}(\lambda x))}^k{\sin }^{-1}(\lambda x{)}^k\mathrm{Gamma}(k+1,-i{\sin }^{-1}(\lambda x))+ic{(-i{\sin }^{-1}(\lambda x))}^k{\sin }^{-1}(\lambda x{)}^k\mathrm{Gamma}(k+1,i{\sin }^{-1}(\lambda x))+2\lambda sx{({\sin }^{-1}(\lambda x{)}^2)}^k)}{2\lambda }\}\right\}$$

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

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

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7.7.20.2 [1704] Problem 2

problem number 1704

Added June 26, 2019.

Problem Chapter 7.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcSin[lambda1*x]+b2*ArcSin[lambda2*y]+b3*ArcSin[lambda3*z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde :=  a1*diff(w(x,y,z),x)+ a2*diff(w(x,y,z),y)+ a3*diff(w(x,y,z),z)= b1*arcsin(lambda1*x)+b2*arcsin(lambda2*y)+b3*arcsin(lambda3*z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\frac{\mathit{a}\mathit{1}\mathit{a}\mathit{2}\mathit{a}\mathit{3}\lambda 1\lambda 2\lambda 3\mathit{\text{\_F1}}(\frac{\mathit{a}\mathit{1}y-\mathit{a}\mathit{2}x}{\mathit{a}\mathit{1}},\frac{\mathit{a}\mathit{1}z-\mathit{a}\mathit{3}x}{\mathit{a}\mathit{1}})+\sqrt{-\lambda 1^2{x}^2+1}\mathit{a}\mathit{2}\mathit{a}\mathit{3}\mathit{b}\mathit{1}\lambda 2\lambda 3+(\sqrt{-\lambda 2^2{y}^2+1}\mathit{a}\mathit{1}\mathit{a}\mathit{3}\mathit{b}\mathit{2}\lambda 3+(\mathit{a}\mathit{2}\mathit{a}\mathit{3}\mathit{b}\mathit{1}\lambda 3x\arcsin \left(\lambda 1x\right)+(\mathit{a}\mathit{3}\mathit{b}\mathit{2}\lambda 3y\arcsin \left(\lambda 2y\right)+(\lambda 3z\arcsin \left(\lambda 3z\right)+\sqrt{-\lambda 3^2{z}^2+1})\mathit{a}\mathit{2}\mathit{b}\mathit{3})\mathit{a}\mathit{1})\lambda 2)\lambda 1}{\mathit{a}\mathit{1}\mathit{a}\mathit{2}\mathit{a}\mathit{3}\lambda 1\lambda 2\lambda 3}$$

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7.7.20.3 [1705] Problem 3

problem number 1705

Added June 26, 2019.

Problem Chapter 7.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*ArcSin[lambda*x]^n*ArcSin[beta*z]^k*D[w[x,y,z],z]== s*ArcSin[gamma*x]^m; 
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*arcsin(lambda*x)^n*arcsin(beta*z)^k*diff(w(x,y,z),z)= s*arcsin(gamma*x)^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\int \frac{s\arcsin {\left(\gamma x\right)}^m}{a}dx+\mathit{\text{\_F1}}(\frac{ay-bx}{a},-\frac{2(\frac{(k-1)(\arcsin {\left(\lambda x\right)}^n-\frac{\mathrm{LommelS}1(n+\frac{3}{2},\frac{1}{2},\arcsin \left(\lambda x\right))}{\sqrt{\arcsin \left(\lambda x\right)}})(-{\lambda }^2{x}^2+1)\beta c\lambda x{2}^n{2}^{-n}}{2}+\frac{(\lambda x-1)(\lambda x+1)(k-1)(\arcsin {\left(\lambda x\right)}^n-\frac{\mathrm{LommelS}1(n+\frac{3}{2},\frac{1}{2},\arcsin \left(\lambda x\right))}{\sqrt{\arcsin \left(\lambda x\right)}})\sqrt{-{\lambda }^2{x}^2+1}\beta c{2}^n{2}^{-n}\arcsin \left(\lambda x\right)}{2}+(\lambda x-1)(-\frac{(n+1)(-\arcsin {\left(\beta z\right)}^{-k}\arcsin {\left(\beta z\right)}^{\frac{3}{2}}+\mathrm{LommelS}1(-k+\frac{3}{2},\frac{1}{2},\arcsin \left(\beta z\right))\arcsin \left(\beta z\right))\sqrt{-{\beta }^2{z}^2+1}a{2}^k{2}^{-k}}{2\sqrt{\arcsin \left(\beta z\right)}}+(-\frac{(n+1)akz{2}^k{2}^{-k}\mathrm{LommelS}1(-k+\frac{1}{2},\frac{3}{2},\arcsin \left(\beta z\right))\sqrt{\arcsin \left(\beta z\right)}}{2}+\frac{(k-1)cnx{2}^n{2}^{-n}\mathrm{LommelS}1(n+\frac{1}{2},\frac{3}{2},\arcsin \left(\lambda x\right))\sqrt{\arcsin \left(\lambda x\right)}}{2}+(k-1)cx{2}^n{2}^{-n-1}\arcsin {\left(\lambda x\right)}^n+\frac{(n+1)az{2}^k{2}^{-k}\mathrm{LommelS}1(-k+\frac{3}{2},\frac{1}{2},\arcsin \left(\beta z\right))}{2\sqrt{\arcsin \left(\beta z\right)}}+(n+1)(-\frac{2^k}{2}+2^{k-1})az{2}^{-k}\arcsin {\left(\beta z\right)}^{-k})\beta )(\lambda x+1)\lambda )}{(n+1)({\lambda }^2{x}^2-1)(k-1)\beta c\lambda })$$

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7.7.20.4 [1706] Problem 4

problem number 1706

Added June 26, 2019.

Problem Chapter 7.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*ArcSin[lambda*x]^n*ArcSin[beta*y]^m*ArcSin[gamma*z]^k*D[w[x,y,z],z]== s; 
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*arcsin(lambda*x)^n*arcsin(beta*y)^m*arcsin(gamma*z)^k*diff(w(x,y,z),z)= s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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7.7.20.5 [1707] Problem 5

problem number 1707

Added June 26, 2019.

Problem Chapter 7.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to {\int }_1^z\frac{s{\sin }^{-1}{\left(\frac{\gamma (ia{(-i{\sin }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,-i{\sin }^{-1}(\beta z)){\sin }^{-1}(\beta z{)}^{-k}-ia{(i{\sin }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,i{\sin }^{-1}(\beta z)){\sin }^{-1}(\beta z{)}^{-k}+{\sin }^{-1}(\beta K[1]{)}^{-k}(-ia\mathrm{Gamma}(1-k,-i{\sin }^{-1}(\beta K[1])){(-i{\sin }^{-1}(\beta K[1]))}^k+2\beta cx{\sin }^{-1}(\beta K[1]{)}^k+ia{(i{\sin }^{-1}(\beta K[1]))}^k\mathrm{Gamma}(1-k,i{\sin }^{-1}(\beta K[1]))))}{2\beta c}\right)}^m{\sin }^{-1}(\beta K[1]{)}^{-k}}{c}dK[1]+c_1(-\frac{cx}{a}-\frac{i{\sin }^{-1}(\beta z{)}^{-k}({(-i{\sin }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,-i{\sin }^{-1}(\beta z))-{(i{\sin }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,i{\sin }^{-1}(\beta z)))}{2\beta },\frac{{({\sin }^{-1}(\lambda x{)}^2)}^{-n}(ib{(i{\sin }^{-1}(\lambda x))}^n{\sin }^{-1}(\lambda x{)}^n\mathrm{Gamma}(n+1,-i{\sin }^{-1}(\lambda x))-ib{(-i{\sin }^{-1}(\lambda x))}^n{\sin }^{-1}(\lambda x{)}^n\mathrm{Gamma}(n+1,i{\sin }^{-1}(\lambda x))+2a\lambda y{({\sin }^{-1}(\lambda x{)}^2)}^n)}{2a\lambda })\}\right\}$$ Generates Solve::incnst: Inconsistent or redundant transcendental equation

Maple ✗

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

time expired

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7.7.20.6 [1708] Problem 6

problem number 1708

Added June 26, 2019.

Problem Chapter 7.7.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to c_1(-\frac{cx}{a}-\frac{i{\sin }^{-1}(\beta z{)}^{-k}({(-i{\sin }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,-i{\sin }^{-1}(\beta z))-{(i{\sin }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,i{\sin }^{-1}(\beta z)))}{2\beta },\frac{{({\sin }^{-1}(\lambda x{)}^2)}^{-n}(ib{(i{\sin }^{-1}(\lambda x))}^n{\sin }^{-1}(\lambda x{)}^n\mathrm{Gamma}(n+1,-i{\sin }^{-1}(\lambda x))-ib{(-i{\sin }^{-1}(\lambda x))}^n{\sin }^{-1}(\lambda x{)}^n\mathrm{Gamma}(n+1,i{\sin }^{-1}(\lambda x))+2a\lambda y{({\sin }^{-1}(\lambda x{)}^2)}^n)}{2a\lambda })-\frac{is{\sin }^{-1}(\beta z{)}^{-k}({(-i{\sin }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,-i{\sin }^{-1}(\beta z))-{(i{\sin }^{-1}(\beta z))}^k\mathrm{Gamma}(1-k,i{\sin }^{-1}(\beta z)))}{2\beta c}\}\right\}$$ Generates Solve::incnst: Inconsistent or redundant transcendental equation

Maple ✗

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

time expired

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