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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -b x}{a}, -\frac {\left (a \sqrt {\arcsin \left (\lambda x \right )}\, \left (n +1\right ) \left (-\operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right ) \arcsin \left (\beta z \right )+\arcsin \left (\beta z \right )^{-k +\frac {3}{2}}\right ) \sqrt {-\beta ^{2} z^{2}+1}-\left (-a \sqrt {\arcsin \left (\lambda x \right )}\, z \left (n +1\right ) \operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right )+a \sqrt {\arcsin \left (\lambda x \right )}\, k z \arcsin \left (\beta z \right ) \left (n +1\right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\beta z \right )\right )-\sqrt {\arcsin \left (\beta z \right )}\, c x \left (k -1\right ) \left (\arcsin \left (\lambda x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) n +\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )\right )\right ) \beta \right ) \lambda \sqrt {-\lambda ^{2} x^{2}+1}+\sqrt {\arcsin \left (\beta z \right )}\, c \beta \left (\lambda x -1\right ) \left (\lambda x +1\right ) \left (k -1\right ) \left (\arcsin \left (\lambda x \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )-\arcsin \left (\lambda x \right )^{n +\frac {3}{2}}\right )}{\sqrt {-\lambda ^{2} x^{2}+1}\, \sqrt {\arcsin \left (\lambda x \right )}\, \sqrt {\arcsin \left (\beta z \right )}\, \left (k -1\right ) \beta c \lambda \left (n +1\right )}\right )\]

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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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -x b}{a}, \frac {-\gamma \operatorname {1} \sqrt {\arcsin \left (\gamma \operatorname {1} z \right )}\, c \left (k -1\right ) \int _{}^{x}\arcsin \left (\lambda \textit {\_a} \right )^{n} \arcsin \left (\frac {\beta \left (a y -b \left (x -\textit {\_a} \right )\right )}{a}\right )^{m}d \textit {\_a} +a \left (\left (\operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\gamma \operatorname {1} z \right )\right ) \arcsin \left (\gamma \operatorname {1} z \right )-\arcsin \left (\gamma \operatorname {1} z \right )^{-k +\frac {3}{2}}\right ) \sqrt {-\gamma \operatorname {1}^{2} z^{2}+1}+\gamma \operatorname {1} z \left (\arcsin \left (\gamma \operatorname {1} z \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\gamma \operatorname {1} z \right )\right ) k -\operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\gamma \operatorname {1} z \right )\right )\right )\right )}{\sqrt {\arcsin \left (\gamma \operatorname {1} z \right )}\, \gamma \operatorname {1} c \left (k -1\right )}\right )\]

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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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b \left (-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right ) \arcsin \left (\lambda x \right )+\arcsin \left (\lambda x \right )^{n +\frac {3}{2}}\right ) \sqrt {-\lambda ^{2} x^{2}+1}+\left (-b x \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )-\operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) b n x \arcsin \left (\lambda x \right )+a \sqrt {\arcsin \left (\lambda x \right )}\, y \left (n +1\right )\right ) \lambda }{\sqrt {\arcsin \left (\lambda x \right )}\, \left (n +1\right ) \lambda a}, \frac {-c \left (-\arcsin \left (\beta x \right ) \operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta x \right )\right )+\arcsin \left (\beta x \right )^{k +\frac {3}{2}}\right ) \sqrt {-\beta ^{2} x^{2}+1}+\left (-c x \operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta x \right )\right )-\operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\beta x \right )\right ) c k x \arcsin \left (\beta x \right )+a \sqrt {\arcsin \left (\beta x \right )}\, z \left (k +1\right )\right ) \beta }{\sqrt {\arcsin \left (\beta x \right )}\, \left (k +1\right ) \beta a}\right )\]

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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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b \left (\sqrt {\arcsin \left (\lambda x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )-\arcsin \left (\lambda x \right )^{n +1}\right ) \sqrt {-\lambda ^{2} x^{2}+1}-\lambda \left (-\frac {\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right ) x b}{\sqrt {\arcsin \left (\lambda x \right )}}-\sqrt {\arcsin \left (\lambda x \right )}\, n \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) x b +a y \left (n +1\right )\right )}{a \lambda \left (n +1\right )}, \frac {-\beta \sqrt {\arcsin \left (\beta z \right )}\, c \left (k -1\right ) \int _{}^{y}{\arcsin \left (\lambda \operatorname {RootOf}\left (\arcsin \left (\lambda \textit {\_Z} \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda \textit {\_Z} \right )\right ) \textit {\_Z} b \lambda n -\sqrt {\arcsin \left (\lambda \textit {\_Z} \right )}\, \lambda n b \int \arcsin \left (\lambda x \right )^{n}d x -\sqrt {\arcsin \left (\lambda \textit {\_Z} \right )}\, a \lambda n \textit {\_b} +\sqrt {\arcsin \left (\lambda \textit {\_Z} \right )}\, a \lambda n y -\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, \arcsin \left (\lambda \textit {\_Z} \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda \textit {\_Z} \right )\right ) b -\lambda \sqrt {\arcsin \left (\lambda \textit {\_Z} \right )}\, b \int \arcsin \left (\lambda x \right )^{n}d x -\textit {\_b} a \lambda \sqrt {\arcsin \left (\lambda \textit {\_Z} \right )}+a \lambda \sqrt {\arcsin \left (\lambda \textit {\_Z} \right )}\, y +\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda \textit {\_Z} \right )\right ) \textit {\_Z} b \lambda +b \sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, \arcsin \left (\lambda \textit {\_Z} \right )^{n +\frac {3}{2}}\right )\right )}^{-n}d \textit {\_b} +\left (\left (\arcsin \left (\beta z \right ) \operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right )-\arcsin \left (\beta z \right )^{-k +\frac {3}{2}}\right ) \sqrt {-\beta ^{2} z^{2}+1}+\beta z \left (\arcsin \left (\beta z \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\beta z \right )\right ) k -\operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right )\right )\right ) b}{\sqrt {\arcsin \left (\beta z \right )}\, \left (k -1\right ) \beta c}\right )\]

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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]];
 

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

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 ) = f_{1} \left (\frac {a \left (-\operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right )+\arcsin \left (\lambda y \right )^{-n +\frac {3}{2}}\right ) \sqrt {-\lambda ^{2} y^{2}+1}-\lambda \left (-a y \operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right )+a \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda y \right )\right ) n y \arcsin \left (\lambda y \right )-\sqrt {\arcsin \left (\lambda y \right )}\, b x \left (n -1\right )\right )}{\sqrt {\arcsin \left (\lambda y \right )}\, \left (n -1\right ) \lambda b}, -\frac {\left (b \sqrt {\arcsin \left (\lambda y \right )}\, \left (n -1\right ) \left (-\operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right ) \arcsin \left (\beta z \right )+\arcsin \left (\beta z \right )^{-k +\frac {3}{2}}\right ) \sqrt {-\beta ^{2} z^{2}+1}-\beta \left (\sqrt {\arcsin \left (\beta z \right )}\, c y \left (k -1\right ) \operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right )-b \sqrt {\arcsin \left (\lambda y \right )}\, z \left (n -1\right ) \operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right )-\arcsin \left (\lambda y \right ) \sqrt {\arcsin \left (\beta z \right )}\, c n y \left (k -1\right ) \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda y \right )\right )+b \sqrt {\arcsin \left (\lambda y \right )}\, \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\beta z \right )\right ) k z \arcsin \left (\beta z \right ) \left (n -1\right )\right )\right ) \lambda \sqrt {-\lambda ^{2} y^{2}+1}-\sqrt {\arcsin \left (\beta z \right )}\, c \beta \left (\lambda y -1\right ) \left (\lambda y +1\right ) \left (k -1\right ) \left (\operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right )-\arcsin \left (\lambda y \right )^{-n +\frac {3}{2}}\right )}{\sqrt {\arcsin \left (\beta z \right )}\, \sqrt {-\lambda ^{2} y^{2}+1}\, \sqrt {\arcsin \left (\lambda y \right )}\, \beta \left (k -1\right ) c \lambda \left (n -1\right )}\right )\]

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