6.7.20 7.1

6.7.20.1 [1701] Problem 1

problem number 1701

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {\left (\arcsin (\lambda x)^2\right )^{-k} \left (-i c (i \arcsin (\lambda x))^k \arcsin (\lambda x)^k \Gamma (k+1,-i \arcsin (\lambda x))+i c (-i \arcsin (\lambda x))^k \arcsin (\lambda x)^k \Gamma (k+1,i \arcsin (\lambda x))+2 \lambda s x \left (\arcsin (\lambda x)^2\right )^k\right )}{2 \lambda }\right \}\right \}\]

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

\[w \left (x , y , z\right ) = c \int \arcsin \left (\lambda x \right )^{k}d x +s x +f_{1} \left (-a x +y , -b x +z \right )\]

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6.7.20.2 [1702] Problem 2

problem number 1702

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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = \frac {\sqrt {-\lambda \operatorname {1}^{2} x^{2}+1}\, \operatorname {a2} \operatorname {a3} \operatorname {b1} \lambda \operatorname {2} \lambda \operatorname {3} +\left (\lambda \operatorname {3} \operatorname {a1} \operatorname {a3} \operatorname {b2} \sqrt {-\lambda \operatorname {2}^{2} y^{2}+1}+\left (x \lambda \operatorname {3} \arcsin \left (\lambda \operatorname {1} x \right ) \operatorname {a2} \operatorname {a3} \operatorname {b1} +\operatorname {a1} \left (\lambda \operatorname {3} \arcsin \left (\lambda \operatorname {2} y \right ) y \operatorname {a3} \operatorname {b2} +\left (\lambda \operatorname {3} \arcsin \left (\lambda \operatorname {3} z \right ) z \operatorname {b3} +f_{1} \left (\frac {y \operatorname {a1} -x \operatorname {a2}}{\operatorname {a1}}, \frac {z \operatorname {a1} -x \operatorname {a3}}{\operatorname {a1}}\right ) \lambda \operatorname {3} \operatorname {a3} +\sqrt {-\lambda \operatorname {3}^{2} z^{2}+1}\, \operatorname {b3} \right ) \operatorname {a2} \right )\right ) \lambda \operatorname {2} \right ) \lambda \operatorname {1} }{\lambda \operatorname {1} \operatorname {a1} \operatorname {a2} \lambda \operatorname {2} \operatorname {a3} \lambda \operatorname {3} }\]

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6.7.20.3 [1703] Problem 3

problem number 1703

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'));
 

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

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6.7.20.4 [1704] Problem 4

problem number 1704

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'));
 

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

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6.7.20.5 [1705] Problem 5

problem number 1705

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]];
 

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

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'));
 

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

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6.7.20.6 [1706] Problem 6

problem number 1706

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x}{a}-\frac {i \arcsin (\beta z)^{-k} \left ((-i \arcsin (\beta z))^k \Gamma (1-k,-i \arcsin (\beta z))-(i \arcsin (\beta z))^k \Gamma (1-k,i \arcsin (\beta z))\right )}{2 \beta },\frac {\left (\arcsin (\lambda x)^2\right )^{-n} \left (2 a \lambda y \left (\arcsin (\lambda x)^2\right )^n+i b (i \arcsin (\lambda x))^n \arcsin (\lambda x)^n \Gamma (n+1,-i \arcsin (\lambda x))-i b (-i \arcsin (\lambda x))^n \arcsin (\lambda x)^n \Gamma (n+1,i \arcsin (\lambda x))\right )}{2 a \lambda }\right )-\frac {i s \arcsin (\beta z)^{-k} \left ((-i \arcsin (\beta z))^k \Gamma (1-k,-i \arcsin (\beta z))-(i \arcsin (\beta z))^k \Gamma (1-k,i \arcsin (\beta z))\right )}{2 \beta c}\right \}\right \}\]

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'));
 

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

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