6.6.19 7.1

6.6.19.1 [1530] Problem 1

problem number 1530

Added May 31, 2019.

Problem Chapter 6.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \arcsin ^n(\lambda x) \arcsin ^k(\beta z) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, -\frac {2 \left (\frac {\left (k -1\right ) \left (\arcsin \left (\lambda x \right )^{n}-\frac {\LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )}{\sqrt {\arcsin \left (\lambda x \right )}}\right ) \left (-\lambda ^{2} x^{2}+1\right ) \beta c \lambda x 2^{n} 2^{-n}}{2}+\frac {\left (\lambda x -1\right ) \left (\lambda x +1\right ) \left (k -1\right ) \left (\arcsin \left (\lambda x \right )^{n}-\frac {\LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )}{\sqrt {\arcsin \left (\lambda x \right )}}\right ) \sqrt {-\lambda ^{2} x^{2}+1}\, \beta c 2^{n} 2^{-n} \arcsin \left (\lambda x \right )}{2}+\left (\lambda x +1\right ) \left (-\frac {\left (n +1\right ) \left (-\arcsin \left (\beta z \right )^{-k} \arcsin \left (\beta z \right )^{\frac {3}{2}}+\LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right ) \arcsin \left (\beta z \right )\right ) \sqrt {-\beta ^{2} z^{2}+1}\, a 2^{k} 2^{-k}}{2 \sqrt {\arcsin \left (\beta z \right )}}+\left (-\frac {\left (n +1\right ) a k z 2^{k} 2^{-k} \LommelS 1\left (-k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\beta z \right )\right ) \sqrt {\arcsin \left (\beta z \right )}}{2}+\frac {\left (k -1\right ) c n x 2^{n} 2^{-n} \LommelS 1\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) \sqrt {\arcsin \left (\lambda x \right )}}{2}+\left (k -1\right ) c x 2^{n} 2^{-n -1} \arcsin \left (\lambda x \right )^{n}+\frac {\left (n +1\right ) a z 2^{k} 2^{-k} \LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right )}{2 \sqrt {\arcsin \left (\beta z \right )}}+\left (n +1\right ) \left (-\frac {2^{k}}{2}+2^{k -1}\right ) a z 2^{-k} \arcsin \left (\beta z \right )^{-k}\right ) \beta \right ) \left (\lambda x -1\right ) \lambda \right )}{\left (n +1\right ) \left (\lambda ^{2} x^{2}-1\right ) \left (k -1\right ) \beta c \lambda }\right )\]

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6.6.19.2 [1531] Problem 2

problem number 1531

Added May 31, 2019.

Problem Chapter 6.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \arcsin ^n(\lambda x) \arcsin ^m(\beta y) \arcsin ^k(\gamma z) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

restart; 
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(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'));
 

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

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6.6.19.3 [1532] Problem 3

problem number 1532

Added May 31, 2019.

Problem Chapter 6.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arcsin ^n(\lambda x) w_y + c \arcsin ^k(\beta x) w_z = 0 \]

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*x]^k*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*arcsin(lambda*x)^n*diff(w(x,y,z),y)+c*arcsin(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'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {\left (-\arcsin \left (\lambda x \right )^{n} \arcsin \left (\lambda x \right )^{\frac {3}{2}}+\LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right ) \arcsin \left (\lambda x \right )\right ) \sqrt {-\lambda ^{2} x^{2}+1}\, b 2^{n} 2^{-n}+\left (-b n x 2^{n} 2^{-n} \LommelS 1\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) \arcsin \left (\lambda x \right )-b x 2^{n} 2^{-n} \LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )+\left (b x 2^{n} 2^{-n} \arcsin \left (\lambda x \right )^{n}-2 b x 2^{n} 2^{-n -1} \arcsin \left (\lambda x \right )^{n}+\left (n +1\right ) a y \right ) \sqrt {\arcsin \left (\lambda x \right )}\right ) \lambda }{\left (n +1\right ) a \lambda \sqrt {\arcsin \left (\lambda x \right )}}, \frac {\left (-\arcsin \left (\beta x \right )^{k} \arcsin \left (\beta x \right )^{\frac {3}{2}}+\LommelS 1\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta x \right )\right ) \arcsin \left (\beta x \right )\right ) \sqrt {-\beta ^{2} x^{2}+1}\, c 2^{k} 2^{-k}+\left (-c k x 2^{k} 2^{-k} \LommelS 1\left (k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\beta x \right )\right ) \arcsin \left (\beta x \right )-c x 2^{k} 2^{-k} \LommelS 1\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta x \right )\right )+\left (c x 2^{k} 2^{-k} \arcsin \left (\beta x \right )^{k}-2 c x 2^{k} 2^{-k -1} \arcsin \left (\beta x \right )^{k}+\left (k +1\right ) a z \right ) \sqrt {\arcsin \left (\beta x \right )}\right ) \beta }{\left (k +1\right ) a \beta \sqrt {\arcsin \left (\beta x \right )}}\right )\]

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6.6.19.4 [1533] Problem 4

problem number 1533

Added May 31, 2019.

Problem Chapter 6.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arcsin ^n(\lambda x) w_y + c \arcsin ^k(\beta z) w_z = 0 \]

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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✗

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

time expired

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6.6.19.5 [1534] Problem 5

problem number 1534

Added May 31, 2019.

Problem Chapter 6.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arcsin ^n(\lambda y) w_y + c \arcsin ^k(\beta z) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*arcsin(lambda*y)^n*diff(w(x,y,z),y)+c*arcsin(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 \left (x , y , z\right ) = \mathit {\_F1} \left (-\frac {\left (-\arcsin \left (\lambda y \right )^{-n} \arcsin \left (\lambda y \right )^{\frac {3}{2}}+\LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right )\right ) \sqrt {-\lambda ^{2} y^{2}+1}\, a 2^{n} 2^{-n}+\left (a n y 2^{n} 2^{-n} \LommelS 1\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right )-a y 2^{n} 2^{-n} \LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right )+\left (\left (2^{n}-2 2^{n -1}\right ) a y 2^{-n} \arcsin \left (\lambda y \right )^{-n}-\left (n -1\right ) b x \right ) \sqrt {\arcsin \left (\lambda y \right )}\right ) \lambda }{\left (n -1\right ) b \lambda \sqrt {\arcsin \left (\lambda y \right )}}, -\frac {2 \left (-\frac {\left (k -1\right ) \left (\arcsin \left (\lambda y \right )^{-n}-\frac {\LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right )}{\sqrt {\arcsin \left (\lambda y \right )}}\right ) \left (-\lambda ^{2} y^{2}+1\right ) \beta c \lambda y 2^{n} 2^{-n}}{2}-\frac {\left (\lambda y -1\right ) \left (\lambda y +1\right ) \left (k -1\right ) \left (\arcsin \left (\lambda y \right )^{-n}-\frac {\LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right )}{\sqrt {\arcsin \left (\lambda y \right )}}\right ) \sqrt {-\lambda ^{2} y^{2}+1}\, \beta c 2^{n} 2^{-n} \arcsin \left (\lambda y \right )}{2}+\left (\lambda y +1\right ) \left (-\frac {\left (n -1\right ) \left (-\arcsin \left (\beta z \right )^{-k} \arcsin \left (\beta z \right )^{\frac {3}{2}}+\LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right ) \arcsin \left (\beta z \right )\right ) \sqrt {-\beta ^{2} z^{2}+1}\, b 2^{k} 2^{-k}}{2 \sqrt {\arcsin \left (\beta z \right )}}+\left (-\frac {\left (n -1\right ) b k z 2^{k} 2^{-k} \LommelS 1\left (-k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\beta z \right )\right ) \sqrt {\arcsin \left (\beta z \right )}}{2}+\frac {\left (k -1\right ) c n y 2^{n} 2^{-n} \LommelS 1\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda y \right )\right ) \sqrt {\arcsin \left (\lambda y \right )}}{2}-\left (k -1\right ) c y 2^{-n} 2^{n -1} \arcsin \left (\lambda y \right )^{-n}+\frac {\left (n -1\right ) b z 2^{k} 2^{-k} \LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right )}{2 \sqrt {\arcsin \left (\beta z \right )}}+\left (n -1\right ) \left (-\frac {2^{k}}{2}+2^{k -1}\right ) b z 2^{-k} \arcsin \left (\beta z \right )^{-k}\right ) \beta \right ) \left (\lambda y -1\right ) \lambda \right )}{\left (n -1\right ) \left (\lambda ^{2} y^{2}-1\right ) \left (k -1\right ) \beta c \lambda }\right )\]

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