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]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {\left (\cos ^{-1}(\lambda x)^2\right )^{-k} \left (c \left (i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text {Gamma}\left (k+1,-i \cos ^{-1}(\lambda x)\right )+c \left (-i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text {Gamma}\left (k+1,i \cos ^{-1}(\lambda x)\right )+2 \lambda s x \left (\cos ^{-1}(\lambda x)^2\right )^k\right )}{2 \lambda }\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \arccos \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \] 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'));
\[w \left ( x,y,z \right ) ={\frac {1}{\lambda 1\,\lambda 2\,{\it a2}\,{\it a1}\,\lambda 3\,{\it a3}} \left ( {\it \_F1} \left ( {\frac {y{\it a1}-{\it a2}\,x}{{\it a1}}},{\frac {z{\it a1}-{\it a3}\,x}{{\it a1}}} \right ) {\it a1}\,\lambda 1\,{\it a2}\,\lambda 2\,{\it a3}\,\lambda 3-\sqrt {-{\lambda 1}^{2}{x}^{2}+1}{\it a2}\,{\it a3}\,{\it b1}\,\lambda 2\,\lambda 3-\lambda 1\, \left ( \lambda 3\,{\it a1}\,{\it a3}\,{\it b2}\,\sqrt {-{\lambda 2}^{2}{y}^{2}+1}+ \left ( -x\lambda 3\,\arccos \left ( \lambda 1\,x \right ) {\it a2}\,{\it a3}\,{\it b1}+{\it a1}\, \left ( -\lambda 3\,\arccos \left ( \lambda 2\,y \right ) y{\it a3}\,{\it b2}+{\it b3}\, \left ( -\lambda 3\,\arccos \left ( \lambda 3\,z \right ) z+\sqrt {-{\lambda 3}^{2}{z}^{2}+1} \right ) {\it a2} \right ) \right ) \lambda 2 \right ) \right ) }\]
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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'));
\[w \left ( x,y,z \right ) =\int \!{\frac {s \left ( \arccos \left ( x\gamma \right ) \right ) ^{m}}{a}}\,{\rm d}x+{\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {{2}^{n}\sqrt {\pi }}{\lambda } \left ( {\frac { \left ( \arccos \left ( x\lambda \right ) \right ) ^{1+n}{2}^{-n}}{\sqrt {\pi } \left ( 2+n \right ) }\sqrt {-{x}^{2}{\lambda }^{2}+1}}-{\frac {{2}^{-n}}{\sqrt {\pi } \left ( 2+n \right ) }\sqrt {\arccos \left ( x\lambda \right ) }\LommelS 1 \left ( n+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( x\lambda \right ) \right ) \sqrt {-{x}^{2}{\lambda }^{2}+1}}-3\,{\frac {{2}^{-n-1} \left ( 4/3+2/3\,n \right ) \left ( \lambda \,x\arccos \left ( x\lambda \right ) -\sqrt {-{x}^{2}{\lambda }^{2}+1} \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x\lambda \right ) \right ) }{\sqrt {\pi } \left ( 2+n \right ) \sqrt {\arccos \left ( x\lambda \right ) }}} \right ) }+{\frac {a{2}^{-k}}{ \left ( k-2 \right ) \beta \,c} \left ( 2\,\arccos \left ( \beta \,z \right ) z{2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) \beta \,k-4\,\arccos \left ( \beta \,z \right ) z{2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) \beta -{2}^{k}\arccos \left ( \beta \,z \right ) \LommelS 1 \left ( -k+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( \beta \,z \right ) \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}-2\,{2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) k+ \left ( \arccos \left ( \beta \,z \right ) \right ) ^{-k+1}{2}^{k}\sqrt {-{\beta }^{2}{z}^{2}+1}\sqrt {\arccos \left ( \beta \,z \right ) }+4\,{2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) \right ) {\frac {1}{\sqrt {\arccos \left ( \beta \,z \right ) }}}} \right ) \]
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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]];
Failed
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}{b} \left ( \arccos \left ( \RootOf \left ( \arccos \left ( {\it \_Z}\,\lambda \right ) \LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) {\it \_Z}\,b\lambda \,n+2\,\arccos \left ( {\it \_Z}\,\lambda \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) {\it \_Z}\,b\lambda -a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }n\int \!{\frac {b \left ( \arccos \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x-a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }n{\it \_a}+a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }ny+\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\arccos \left ( {\it \_Z}\,\lambda \right ) \LommelS 1 \left ( n+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) b-\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) bn-2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }\int \!{\frac {b \left ( \arccos \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x-2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }{\it \_a}+2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }y-2\,\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) b- \left ( \arccos \left ( {\it \_Z}\,\lambda \right ) \right ) ^{n+{\frac {3}{2}}}\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b \right ) \gamma \right ) \right ) ^{m} \left ( \arccos \left ( \RootOf \left ( \arccos \left ( {\it \_Z}\,\lambda \right ) \LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) {\it \_Z}\,b\lambda \,n+2\,\arccos \left ( {\it \_Z}\,\lambda \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) {\it \_Z}\,b\lambda -a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }n\int \!{\frac {b \left ( \arccos \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x-a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }n{\it \_a}+a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }ny+\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\arccos \left ( {\it \_Z}\,\lambda \right ) \LommelS 1 \left ( n+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) b-\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) bn-2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }\int \!{\frac {b \left ( \arccos \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x-2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }{\it \_a}+2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }y-2\,\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) b- \left ( \arccos \left ( {\it \_Z}\,\lambda \right ) \right ) ^{n+{\frac {3}{2}}}\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b \right ) \lambda \right ) \right ) ^{-n}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {1}{ \left ( 2+n \right ) a\lambda } \left ( -{{2}^{n}b{2}^{-n} \left ( \left ( 2+n \right ) \LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( x\lambda \right ) \right ) -\arccos \left ( x\lambda \right ) \LommelS 1 \left ( n+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( x\lambda \right ) \right ) + \left ( \arccos \left ( x\lambda \right ) \right ) ^{n+{\frac {3}{2}}} \right ) \sqrt {-{x}^{2}{\lambda }^{2}+1}{\frac {1}{\sqrt {\arccos \left ( x\lambda \right ) }}}}-\lambda \, \left ( 2+n \right ) \left ( -\sqrt {\arccos \left ( x\lambda \right ) }bx{2}^{n}{2}^{-n}\LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( x\lambda \right ) \right ) +ay \right ) \right ) },{\frac {1}{ \left ( k-2 \right ) \beta \,c} \left ( -\int ^{y}\! \left ( \arccos \left ( \RootOf \left ( \arccos \left ( {\it \_Z}\,\lambda \right ) \LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) {\it \_Z}\,b\lambda \,n-a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }n\int \!{\frac {b \left ( \arccos \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+2\,\arccos \left ( {\it \_Z}\,\lambda \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) {\it \_Z}\,b\lambda -{\it \_b}\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }n+a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }ny-2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }\int \!{\frac {b \left ( \arccos \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\arccos \left ( {\it \_Z}\,\lambda \right ) \LommelS 1 \left ( n+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) b-\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) bn-2\,{\it \_b}\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }+2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }y- \left ( \arccos \left ( {\it \_Z}\,\lambda \right ) \right ) ^{n+{\frac {3}{2}}}\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b-2\,\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) b \right ) \lambda \right ) \right ) ^{-n}{d{\it \_b}}c\beta \, \left ( k-2 \right ) +{b{2}^{-k}{2}^{k} \left ( \left ( \left ( -k+2 \right ) \LommelS 1 \left ( -k+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( \beta \,z \right ) \right ) -\arccos \left ( \beta \,z \right ) \LommelS 1 \left ( -k+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( \beta \,z \right ) \right ) + \left ( \arccos \left ( \beta \,z \right ) \right ) ^{-k+{\frac {3}{2}}} \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}+\beta \,\arccos \left ( \beta \,z \right ) \LommelS 1 \left ( -k+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( \beta \,z \right ) \right ) z \left ( k-2 \right ) \right ) {\frac {1}{\sqrt {\arccos \left ( \beta \,z \right ) }}}} \right ) } \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 \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n}}{b}}\,{\rm d}y+{\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 ( -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 ( {2}^{k-1}-1/2\,{2}^{k} \right ) b{2}^{-k} \left ( n-1 \right ) -{2}^{-n}{2}^{n-1}c \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n}y \left ( k-1 \right ) \right ) \right ) \left ( \lambda \,y+1 \right ) \lambda \right ) } \right ) \]
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