Added June 26, 2019.
Problem Chapter 7.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \arccos ^k(\lambda x)+s \]
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*ArcCos[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {\left (\cos ^{-1}(\lambda x)^2\right )^{-k} \left (c \left (i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \operatorname {Gamma}\left (k+1,-i \cos ^{-1}(\lambda x)\right )+c \left (-i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \operatorname {Gamma}\left (k+1,i \cos ^{-1}(\lambda x)\right )+2 \lambda s x \left (\cos ^{-1}(\lambda x)^2\right )^k\right )}{2 \lambda }\right \}\right \}\]
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*arccos(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = s x +\int c \arccos \left (\lambda x \right )^{k}d x +\textit {\_F1} \left (-a x +y , -b x +z \right )\] Answer contains unresolved integrals
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Added June 26, 2019.
Problem Chapter 7.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 w_x + a_2 w_y + a_3 w_z = b_1 \arccos (\lambda _1 x)+b_2 \arccos (\lambda _2 y)+b_3 \arccos (\lambda _3 z) \]
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*ArcCos[lambda1*x]+b2*ArcCos[lambda2*y]+b3*ArcCos[lambda3*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right )-\frac {\text {b1} \sqrt {1-\text {lambda1}^2 x^2}}{\text {a1} \text {lambda1}}+\frac {\text {b1} x \cos ^{-1}(\text {lambda1} x)}{\text {a1}}+\frac {\text {b2} x \sin ^{-1}(\text {lambda2} y)}{\text {a1}}+\frac {\text {b2} x \cos ^{-1}(\text {lambda2} y)}{\text {a1}}+\frac {\text {b3} x \sin ^{-1}(\text {lambda3} z)}{\text {a1}}+\frac {\text {b3} x \cos ^{-1}(\text {lambda3} z)}{\text {a1}}-\frac {\text {b2} \sqrt {1-\text {lambda2}^2 y^2}}{\text {a2} \text {lambda2}}-\frac {\text {b2} y \sin ^{-1}(\text {lambda2} y)}{\text {a2}}-\frac {\text {b3} \sqrt {1-\text {lambda3}^2 z^2}}{\text {a3} \text {lambda3}}-\frac {\text {b3} z \sin ^{-1}(\text {lambda3} z)}{\text {a3}}\right \}\right \}\]
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*arccos(lambda1*x)+b2*arccos(lambda2*y)+b3*arccos(lambda3*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\mathit {a1} \mathit {a2} \mathit {a3} \lambda 1 \lambda 2 \lambda 3 \textit {\_F1} \left (\frac {\mathit {a1} y -\mathit {a2} x}{\mathit {a1}}, \frac {\mathit {a1} z -\mathit {a3} x}{\mathit {a1}}\right )-\sqrt {-\lambda 1^{2} x^{2}+1}\, \mathit {a2} \mathit {a3} \mathit {b1} \lambda 2 \lambda 3 -\left (\sqrt {-\lambda 2^{2} y^{2}+1}\, \mathit {a1} \mathit {a3} \mathit {b2} \lambda 3 +\left (-\mathit {a2} \mathit {a3} \mathit {b1} \lambda 3 x \arccos \left (\lambda 1 x \right )+\left (-\mathit {a3} \mathit {b2} \lambda 3 y \arccos \left (\lambda 2 y \right )+\left (-\lambda 3 z \arccos \left (\lambda 3 z \right )+\sqrt {-\lambda 3^{2} z^{2}+1}\right ) \mathit {a2} \mathit {b3} \right ) \mathit {a1} \right ) \lambda 2 \right ) \lambda 1}{\mathit {a1} \mathit {a2} \mathit {a3} \lambda 1 \lambda 2 \lambda 3}\]
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Added June 26, 2019.
Problem Chapter 7.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arccos ^n(\lambda x) \arccos ^k(\beta z) w_z = s \arccos ^m(\gamma x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*ArcCos[lambda*x]^n*ArcCos[beta*z]^k*D[w[x,y,z],z]== s*ArcCos[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*arccos(lambda*x)^n*arccos(beta*z)^k*diff(w(x,y,z),z)= s*arccos(gamma*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int \frac {s \arccos \left (\gamma x \right )^{m}}{a}d x +\textit {\_F1} \left (\frac {a y -b x}{a}, \frac {\sqrt {\pi }\, \left (-\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-n} \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}}{\sqrt {\pi }\, \left (n +2\right )}+\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-n} \arccos \left (\lambda x \right )^{n +1}}{\sqrt {\pi }\, \left (n +2\right )}-\frac {3 \left (\frac {2 n}{3}+\frac {4}{3}\right ) \left (\lambda x \arccos \left (\lambda x \right )-\sqrt {-\lambda ^{2} x^{2}+1}\right ) 2^{-n -1} \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )}{\sqrt {\pi }\, \left (n +2\right ) \sqrt {\arccos \left (\lambda x \right )}}\right ) 2^{n}}{\lambda }+\frac {\left (2 \beta k z 2^{k -1} \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )-4 \beta z 2^{k -1} \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )-2 \sqrt {-\beta ^{2} z^{2}+1}\, k 2^{k -1} \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )-\sqrt {-\beta ^{2} z^{2}+1}\, 2^{k} \LommelS 1 \left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\sqrt {-\beta ^{2} z^{2}+1}\, 2^{k} \arccos \left (\beta z \right )^{-k +1} \sqrt {\arccos \left (\beta z \right )}+4 \sqrt {-\beta ^{2} z^{2}+1}\, 2^{k -1} \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )\right ) a 2^{-k}}{\left (k -2\right ) \beta c \sqrt {\arccos \left (\beta z \right )}}\right )\]
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Added June 26, 2019.
Problem Chapter 7.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arccos ^n(\lambda x) w_y + c \arccos ^k(\beta z) w_z = s \arccos ^m(\gamma x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*ArcCos[lambda*x]^n*D[w[x, y,z], y] + c*ArcCos[beta*z]^k*D[w[x,y,z],z]== s*ArcCos[gamma*x]^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^z\frac {s \cos ^{-1}\left (\frac {\gamma \left (-a \left (-i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right ) \cos ^{-1}(\beta z)^{-k}-a \left (i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right ) \cos ^{-1}(\beta z)^{-k}+\cos ^{-1}(\beta K[1])^{-k} \left (a \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\beta K[1])\right ) \left (-i \cos ^{-1}(\beta K[1])\right )^k+2 \beta c x \cos ^{-1}(\beta K[1])^k+a \left (i \cos ^{-1}(\beta K[1])\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\beta K[1])\right )\right )\right )}{2 \beta c}\right )^m \cos ^{-1}(\beta K[1])^{-k}}{c}dK[1]+c_1\left (-\frac {c x}{a}+\frac {\cos ^{-1}(\beta z)^{-k} \left (\left (-i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right )+\left (i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right )\right )}{2 \beta },\frac {\left (\cos ^{-1}(\lambda x)^2\right )^{-n} \left (-b \left (i \cos ^{-1}(\lambda x)\right )^n \cos ^{-1}(\lambda x)^n \operatorname {Gamma}\left (n+1,-i \cos ^{-1}(\lambda x)\right )-b \left (-i \cos ^{-1}(\lambda x)\right )^n \cos ^{-1}(\lambda x)^n \operatorname {Gamma}\left (n+1,i \cos ^{-1}(\lambda x)\right )+2 a \lambda y \left (\cos ^{-1}(\lambda x)^2\right )^n\right )}{2 a \lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*arccos(lambda*x)^n*diff(w(x,y,z),y)+ c*arccos(beta*z)^k*diff(w(x,y,z),z)= s*arccos(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 ) = \int _{}^{y}\frac {s \arccos \left (\gamma \RootOf \left (\textit {\_Z} b \lambda n \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )+2 \textit {\_Z} b \lambda \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )-\textit {\_a} a \lambda n \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}+a \lambda n y \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-a \lambda n \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-2 \textit {\_a} a \lambda \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}+2 a \lambda y \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-2 a \lambda \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b n \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right )+\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )-\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \arccos \left (\textit {\_Z} \lambda \right )^{n +\frac {3}{2}}-2 \sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right )\right )\right )^{m} \arccos \left (\lambda \RootOf \left (\textit {\_Z} b \lambda n \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )+2 \textit {\_Z} b \lambda \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )-\textit {\_a} a \lambda n \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}+a \lambda n y \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-a \lambda n \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-2 \textit {\_a} a \lambda \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}+2 a \lambda y \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-2 a \lambda \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b n \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right )+\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )-\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \arccos \left (\textit {\_Z} \lambda \right )^{n +\frac {3}{2}}-2 \sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right )\right )\right )^{-n}}{b}d \textit {\_a} +\textit {\_F1} \left (\frac {-\frac {\left (-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \arccos \left (\lambda x \right )+\arccos \left (\lambda x \right )^{n +\frac {3}{2}}+\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )\right ) \sqrt {-\lambda ^{2} x^{2}+1}\, b 2^{n} 2^{-n}}{\sqrt {\arccos \left (\lambda x \right )}}-\left (n +2\right ) \left (-b x 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}+a y \right ) \lambda }{\left (n +2\right ) a \lambda }, \frac {-\left (k -2\right ) \beta c \left (\int _{}^{y}\arccos \left (\lambda \RootOf \left (\textit {\_Z} b \lambda n \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )+2 \textit {\_Z} b \lambda \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )-\textit {\_b} a \lambda n \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}+a \lambda n y \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-a \lambda n \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-2 \textit {\_b} a \lambda \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}+2 a \lambda y \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-2 a \lambda \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b n \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right )+\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )-\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \arccos \left (\textit {\_Z} \lambda \right )^{n +\frac {3}{2}}-2 \sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right )\right )\right )^{-n}d \textit {\_b} \right )+\frac {\left (\left (k -2\right ) \beta z \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\left (-\LommelS 1 \left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\arccos \left (\beta z \right )^{-k +\frac {3}{2}}+\left (-k +2\right ) \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )\right ) \sqrt {-\beta ^{2} z^{2}+1}\right ) b 2^{k} 2^{-k}}{\sqrt {\arccos \left (\beta z \right )}}}{\left (k -2\right ) \beta c}\right )\]
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Added June 26, 2019.
Problem Chapter 7.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arcsin ^n(\lambda y) w_y + c \arcsin ^k(\beta z) w_z = s \]
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]== 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*arcsin(lambda*y)^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 ) = \int \frac {s \arcsin \left (\lambda y \right )^{-n}}{b}d y +\textit {\_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 (\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 (-\frac {2^{k}}{2}+2^{k -1}\right ) \left (n -1\right ) b z 2^{-k} \arcsin \left (\beta z \right )^{-k}\right ) \beta \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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