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]];
 

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

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 ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},-2\,{\frac {1}{ \left ( 1+n \right ) \lambda \, \left ( {x}^{2}{\lambda }^{2}-1 \right ) c\beta \, \left ( k-1 \right ) } \left ( 1/2\,{2}^{n}\lambda \,{2}^{-n}cx\beta \, \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x\lambda \right ) \right ) }{\sqrt {\arcsin \left ( x\lambda \right ) }}}+ \left ( \arcsin \left ( x\lambda \right ) \right ) ^{n} \right ) \left ( -{x}^{2}{\lambda }^{2}+1 \right ) +1/2\,{2}^{n}{2}^{-n}\arcsin \left ( x\lambda \right ) c\beta \, \left ( x\lambda -1 \right ) \left ( x\lambda +1 \right ) \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x\lambda \right ) \right ) }{\sqrt {\arcsin \left ( x\lambda \right ) }}}+ \left ( \arcsin \left ( x\lambda \right ) \right ) ^{n} \right ) \sqrt {-{x}^{2}{\lambda }^{2}+1}+ \left ( x\lambda -1 \right ) \left ( x\lambda +1 \right ) \left ( -1/2\,{\frac {a{2}^{-k}{2}^{k} \left ( 1+n \right ) \left ( \arcsin \left ( \beta \,z \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) - \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k} \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{3/2} \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}}{\sqrt {\arcsin \left ( \beta \,z \right ) }}}+\beta \, \left ( 1/2\,{\frac {a{2}^{-k}z{2}^{k} \left ( 1+n \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) }{\sqrt {\arcsin \left ( \beta \,z \right ) }}}-1/2\,a{2}^{-k}\sqrt {\arcsin \left ( \beta \,z \right ) }kz{2}^{k} \left ( 1+n \right ) \LommelS 1 \left ( -k+1/2,3/2,\arcsin \left ( \beta \,z \right ) \right ) +1/2\,{2}^{n}{2}^{-n}c\sqrt {\arcsin \left ( x\lambda \right ) }nx \left ( k-1 \right ) \LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x\lambda \right ) \right ) +z \left ( 1+n \right ) \left ( {2}^{k-1}-1/2\,{2}^{k} \right ) a{2}^{-k} \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k}+{2}^{n}{2}^{-n-1}c \left ( \arcsin \left ( x\lambda \right ) \right ) ^{n}x \left ( k-1 \right ) \right ) \right ) \lambda \right ) } \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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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]];
 

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

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 ) ={\it \_F1} \left ( -{\frac {1}{ \left ( n-1 \right ) b\lambda } \left ( a{2}^{n}{2}^{-n} \left ( \arcsin \left ( \lambda \,y \right ) \LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,y \right ) \right ) - \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n} \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{{\frac {3}{2}}} \right ) \sqrt {-{\lambda }^{2}{y}^{2}+1}+ \left ( -ay{2}^{n}{2}^{-n}\LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,y \right ) \right ) +a\arcsin \left ( \lambda \,y \right ) \LommelS 1 \left ( -n+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \lambda \,y \right ) \right ) ny{2}^{n}{2}^{-n}+ \left ( ay{2}^{-n} \left ( {2}^{n}-2\,{2}^{n-1} \right ) \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n}-xb \left ( n-1 \right ) \right ) \sqrt {\arcsin \left ( \lambda \,y \right ) } \right ) \lambda \right ) {\frac {1}{\sqrt {\arcsin \left ( \lambda \,y \right ) }}}},-2\,{\frac {1}{\lambda \, \left ( n-1 \right ) \left ( {\lambda }^{2}{y}^{2}-1 \right ) c\beta \, \left ( k-1 \right ) } \left ( -1/2\,{2}^{n}\lambda \,{2}^{-n}cy\beta \, \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( \lambda \,y \right ) \right ) }{\sqrt {\arcsin \left ( \lambda \,y \right ) }}}+ \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n} \right ) \left ( -{\lambda }^{2}{y}^{2}+1 \right ) -1/2\,{2}^{n}{2}^{-n}\arcsin \left ( \lambda \,y \right ) c\beta \, \left ( \lambda \,y-1 \right ) \left ( \lambda \,y+1 \right ) \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( \lambda \,y \right ) \right ) }{\sqrt {\arcsin \left ( \lambda \,y \right ) }}}+ \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n} \right ) \sqrt {-{\lambda }^{2}{y}^{2}+1}+ \left ( \lambda \,y-1 \right ) \left ( \lambda \,y+1 \right ) \left ( -1/2\,{\frac {b{2}^{-k}{2}^{k} \left ( n-1 \right ) \left ( \arcsin \left ( \beta \,z \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) - \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k} \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{3/2} \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}}{\sqrt {\arcsin \left ( \beta \,z \right ) }}}+\beta \, \left ( 1/2\,{\frac {b{2}^{-k}z{2}^{k} \left ( n-1 \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) }{\sqrt {\arcsin \left ( \beta \,z \right ) }}}+1/2\,{2}^{n}{2}^{-n}c\sqrt {\arcsin \left ( \lambda \,y \right ) }ny \left ( k-1 \right ) \LommelS 1 \left ( -n+1/2,3/2,\arcsin \left ( \lambda \,y \right ) \right ) -1/2\,b{2}^{-k}\sqrt {\arcsin \left ( \beta \,z \right ) }kz{2}^{k} \left ( n-1 \right ) \LommelS 1 \left ( -k+1/2,3/2,\arcsin \left ( \beta \,z \right ) \right ) +z \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k} \left ( n-1 \right ) \left ( {2}^{k-1}-1/2\,{2}^{k} \right ) b{2}^{-k}-{2}^{-n}{2}^{n-1}c \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n}y \left ( k-1 \right ) \right ) \right ) \lambda \right ) } \right ) \]

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