Added April 13, 2019.
Problem Chapter 5.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \arccos ^k(\lambda x) + c_2 \arccos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcCos[lambda*x]^k+c2*ArcCos[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \cos ^{-1}(\lambda K[1])^k+\text {c2} \cos ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*arccos(lambda*x)^k+c2*arccos(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {{\it \_a}}{a}}}} \left ( {\it c2}\, \left ( \arccos \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \beta }{a}} \right ) \right ) ^{n}+{\it c1}\, \left ( \arccos \left ( {\it \_a}\,\lambda \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]
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Added April 13, 2019.
Problem Chapter 5.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \arccos ^k(\lambda x) \arccos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ ArcCos[lambda*x]^k*ArcCos[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cos ^{-1}(\lambda K[1])^k \cos ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ arccos(lambda*x)^k*arccos(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \arccos \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \arccos \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \beta }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
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Added April 13, 2019.
Problem Chapter 5.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c_1 \arccos (\lambda _1 x) + c_2 \arccos (\lambda _2 y)\right ) w+ s_1 \arccos ^n(\beta _1 x)+ s_2 \arccos ^k(\beta _2 y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcCos[lambda1*x] + c2*ArcCos[lambda2*y])*w[x,y]+ s1*ArcCos[beta1*x]^n+ s2*ArcCos[beta2*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (-\frac {\text {c1} \sqrt {1-\text {lambda1}^2 x^2}}{a \text {lambda1}}+\frac {\text {c1} x \cos ^{-1}(\text {lambda1} x)}{a}+\frac {\text {c2} x \sin ^{-1}(\text {lambda2} y)}{a}+\frac {\text {c2} x \cos ^{-1}(\text {lambda2} y)}{a}-\frac {\text {c2} \sqrt {1-\text {lambda2}^2 y^2}}{b \text {lambda2}}-\frac {\text {c2} y \sin ^{-1}(\text {lambda2} y)}{b}\right ) \left (\int _1^x\frac {\exp \left (\frac {\text {c2} \text {lambda2} (a y-b x) \sin ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \text {lambda1}-b \text {c1} \text {lambda2} \cos ^{-1}(\text {lambda1} K[1]) K[1] \text {lambda1}-b \text {c2} \text {lambda2} \cos ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) K[1] \text {lambda1}+a \text {c2} \sqrt {-y^2 \text {lambda2}^2-\frac {b^2 (x-K[1])^2 \text {lambda2}^2}{a^2}+\frac {2 b y (x-K[1]) \text {lambda2}^2}{a}+1} \text {lambda1}+b \text {c1} \text {lambda2} \sqrt {1-\text {lambda1}^2 K[1]^2}}{a b \text {lambda1} \text {lambda2}}\right ) \left (\text {s2} \cos ^{-1}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )^k+\text {s1} \cos ^{-1}(\text {beta1} K[1])^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arccos(lambda1*x) + c2*arccos(lambda2*y))*w(x,y)+ s1*arccos(beta1*x)^n+ s2*arccos(beta2*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{{\frac {1}{a\lambda 1\,b\lambda 2} \left ( \sqrt {-{\frac { \left ( \left ( \lambda 2\,y-1 \right ) a-\lambda 2\,b \left ( x-{\it \_a} \right ) \right ) \left ( \left ( \lambda 2\,y+1 \right ) a-\lambda 2\,b \left ( x-{\it \_a} \right ) \right ) }{{a}^{2}}}}a{\it c2}\,\lambda 1- \left ( \lambda 1\,{\it c2}\, \left ( \left ( {\it \_a}-x \right ) b+ya \right ) \arccos \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \lambda 2}{a}} \right ) +{\it c1}\,b \left ( \arccos \left ( \lambda 1\,{\it \_a} \right ) {\it \_a}\,\lambda 1-\sqrt {-{{\it \_a}}^{2}{\lambda 1}^{2}+1} \right ) \right ) \lambda 2 \right ) }}} \left ( {\it s1}\, \left ( \arccos \left ( \beta 1\,{\it \_a} \right ) \right ) ^{n}+{\it s2}\, \left ( \arccos \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \beta 2}{a}} \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {-\sqrt {-{\lambda 2}^{2}{y}^{2}+1}a{\it c2}\,\lambda 1+\lambda 2\, \left ( -b{\it c1}\,\sqrt {-{\lambda 1}^{2}{x}^{2}+1}+\lambda 1\, \left ( a\arccos \left ( \lambda 2\,y \right ) y{\it c2}+bx{\it c1}\,\arccos \left ( \lambda 1\,x \right ) \right ) \right ) }{a\lambda 1\,b\lambda 2}}}}\]
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Added April 13, 2019.
Problem Chapter 5.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccos ^m(\mu x) w_y = c \arccos ^k(\nu x) w + p \arccos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCos[mu*x]^m*D[w[x, y], y] == c*ArcCos[nu*x]^k*w[x,y]+p*ArcCos[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {c \cos ^{-1}(\nu x)^k \left (\cos ^{-1}(\nu x)^2\right )^{-k} \left (\left (-i \cos ^{-1}(\nu x)\right )^k \text {Gamma}\left (k+1,i \cos ^{-1}(\nu x)\right )+\left (i \cos ^{-1}(\nu x)\right )^k \text {Gamma}\left (k+1,-i \cos ^{-1}(\nu x)\right )\right )}{2 a \nu }\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \cos ^{-1}(\nu K[1])^k \left (\cos ^{-1}(\nu K[1])^2\right )^{-k} \left (\text {Gamma}\left (k+1,i \cos ^{-1}(\nu K[1])\right ) \left (-i \cos ^{-1}(\nu K[1])\right )^k+\left (i \cos ^{-1}(\nu K[1])\right )^k \text {Gamma}\left (k+1,-i \cos ^{-1}(\nu K[1])\right )\right )}{2 a \nu }\right ) p \cos ^{-1}\left (\frac {\beta \left (\cos ^{-1}(\mu K[1])^2\right )^{-m} \left (\left (\cos ^{-1}(\mu K[1])^2\right )^m \left (-b \left (i \cos ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,-i \cos ^{-1}(\mu x)\right ) \cos ^{-1}(\mu x)^m-b \left (-i \cos ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,i \cos ^{-1}(\mu x)\right ) \cos ^{-1}(\mu x)^m+2 a \mu y \left (\cos ^{-1}(\mu x)^2\right )^m\right ) \left (\cos ^{-1}(\mu x)^2\right )^{-m}+b \left (i \cos ^{-1}(\mu K[1])\right )^m \cos ^{-1}(\mu K[1])^m \text {Gamma}\left (m+1,-i \cos ^{-1}(\mu K[1])\right )+b \left (-i \cos ^{-1}(\mu K[1])\right )^m \cos ^{-1}(\mu K[1])^m \text {Gamma}\left (m+1,i \cos ^{-1}(\mu K[1])\right )\right )}{2 a \mu }\right )^n}{a}dK[1]+c_1\left (\frac {\left (\cos ^{-1}(\mu x)^2\right )^{-m} \left (-b \left (i \cos ^{-1}(\mu x)\right )^m \cos ^{-1}(\mu x)^m \text {Gamma}\left (m+1,-i \cos ^{-1}(\mu x)\right )-b \left (-i \cos ^{-1}(\mu x)\right )^m \cos ^{-1}(\mu x)^m \text {Gamma}\left (m+1,i \cos ^{-1}(\mu x)\right )+2 a \mu y \left (\cos ^{-1}(\mu x)^2\right )^m\right )}{2 a \mu }\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arccos(mu*x)^m*diff(w(x,y),y) = c*arccos(nu*x)^k*w(x,y)+p*arccos(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p}{a} \left ( \arccos \left ( {\frac {\beta }{\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) } \left ( m+2 \right ) a\mu } \left ( {\frac {b \left ( \left ( m+2 \right ) \LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) \sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }\sqrt {-{\mu }^{2}{x}^{2}+1}}{\sqrt {\arccos \left ( \mu \,x \right ) }}}+ \left ( \left ( -m-2 \right ) \LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) +\LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) \arccos \left ( \mu \,{\it \_a} \right ) - \left ( \arccos \left ( \mu \,{\it \_a} \right ) \right ) ^{m+3/2} \right ) b\sqrt {-{{\it \_a}}^{2}{\mu }^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -\sqrt {\arccos \left ( \mu \,x \right ) }\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }bx\LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) +a\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }y+{\it \_a}\,\LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) \arccos \left ( \mu \,{\it \_a} \right ) b \right ) \right ) } \right ) \right ) ^{n}{{\rm e}^{-{\frac { \left ( \left ( \left ( -2-k \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \nu \,{\it \_a} \right ) \right ) +\arccos \left ( \nu \,{\it \_a} \right ) \LommelS 1 \left ( 3/2+k,3/2,\arccos \left ( \nu \,{\it \_a} \right ) \right ) - \left ( \arccos \left ( \nu \,{\it \_a} \right ) \right ) ^{3/2+k} \right ) \sqrt {-{{\it \_a}}^{2}{\nu }^{2}+1}+\arccos \left ( \nu \,{\it \_a} \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \nu \,{\it \_a} \right ) \right ) \nu \,{\it \_a}\, \left ( 2+k \right ) \right ) c}{\sqrt {\arccos \left ( \nu \,{\it \_a} \right ) }a\nu \, \left ( 2+k \right ) }}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {b \left ( \left ( m+2 \right ) \LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) \sqrt {-{\mu }^{2}{x}^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -bx\LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) \arccos \left ( \mu \,x \right ) +ya\sqrt {\arccos \left ( \mu \,x \right ) } \right ) }{a\mu \, \left ( m+2 \right ) \sqrt {\arccos \left ( \mu \,x \right ) }}} \right ) \right ) {{\rm e}^{-{\frac { \left ( \left ( \left ( 2+k \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \nu \,x \right ) \right ) -\arccos \left ( \nu \,x \right ) \LommelS 1 \left ( 3/2+k,3/2,\arccos \left ( \nu \,x \right ) \right ) + \left ( \arccos \left ( \nu \,x \right ) \right ) ^{3/2+k} \right ) \sqrt {-{\nu }^{2}{x}^{2}+1}-\arccos \left ( \nu \,x \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \nu \,x \right ) \right ) \nu \,x \left ( 2+k \right ) \right ) c{2}^{-k}{2}^{k}}{a\nu \, \left ( 2+k \right ) \sqrt {\arccos \left ( \nu \,x \right ) }}}}}\]
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Added April 13, 2019.
Problem Chapter 5.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccos ^m(\mu x) w_y = c \arccos ^k(\nu y) w + p \arccos ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCos[mu*x]^m*D[w[x, y], y] == c*ArcCos[nu*y]^k*w[x,y]+p*ArcCos[beta*x]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cos ^{-1}\left (\frac {\nu \left (\cos ^{-1}(\mu K[1])^2\right )^{-m} \left (\left (\cos ^{-1}(\mu K[1])^2\right )^m \left (-b \left (i \cos ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,-i \cos ^{-1}(\mu x)\right ) \cos ^{-1}(\mu x)^m-b \left (-i \cos ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,i \cos ^{-1}(\mu x)\right ) \cos ^{-1}(\mu x)^m+2 a \mu y \left (\cos ^{-1}(\mu x)^2\right )^m\right ) \left (\cos ^{-1}(\mu x)^2\right )^{-m}+b \left (i \cos ^{-1}(\mu K[1])\right )^m \cos ^{-1}(\mu K[1])^m \text {Gamma}\left (m+1,-i \cos ^{-1}(\mu K[1])\right )+b \left (-i \cos ^{-1}(\mu K[1])\right )^m \cos ^{-1}(\mu K[1])^m \text {Gamma}\left (m+1,i \cos ^{-1}(\mu K[1])\right )\right )}{2 a \mu }\right )^k}{a}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {c \cos ^{-1}\left (\frac {\nu \left (\cos ^{-1}(\mu K[1])^2\right )^{-m} \left (\left (\cos ^{-1}(\mu K[1])^2\right )^m \left (-b \left (i \cos ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,-i \cos ^{-1}(\mu x)\right ) \cos ^{-1}(\mu x)^m-b \left (-i \cos ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,i \cos ^{-1}(\mu x)\right ) \cos ^{-1}(\mu x)^m+2 a \mu y \left (\cos ^{-1}(\mu x)^2\right )^m\right ) \left (\cos ^{-1}(\mu x)^2\right )^{-m}+b \left (i \cos ^{-1}(\mu K[1])\right )^m \cos ^{-1}(\mu K[1])^m \text {Gamma}\left (m+1,-i \cos ^{-1}(\mu K[1])\right )+b \left (-i \cos ^{-1}(\mu K[1])\right )^m \cos ^{-1}(\mu K[1])^m \text {Gamma}\left (m+1,i \cos ^{-1}(\mu K[1])\right )\right )}{2 a \mu }\right )^k}{a}dK[1]\right ) p \cos ^{-1}(\beta K[2])^n}{a}dK[2]+c_1\left (\frac {\left (\cos ^{-1}(\mu x)^2\right )^{-m} \left (-b \left (i \cos ^{-1}(\mu x)\right )^m \cos ^{-1}(\mu x)^m \text {Gamma}\left (m+1,-i \cos ^{-1}(\mu x)\right )-b \left (-i \cos ^{-1}(\mu x)\right )^m \cos ^{-1}(\mu x)^m \text {Gamma}\left (m+1,i \cos ^{-1}(\mu x)\right )+2 a \mu y \left (\cos ^{-1}(\mu x)^2\right )^m\right )}{2 a \mu }\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arccos(mu*x)^m*diff(w(x,y),y) = c*arccos(nu*y)^k*w(x,y)+p*arccos(beta*x)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p \left ( \arccos \left ( \beta \,{\it \_b} \right ) \right ) ^{n}}{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \arccos \left ( {\frac {\nu }{\sqrt {\arccos \left ( \mu \,{\it \_b} \right ) } \left ( m+2 \right ) a\mu } \left ( {\frac {b \left ( \left ( m+2 \right ) \LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) \sqrt {\arccos \left ( \mu \,{\it \_b} \right ) }\sqrt {-{\mu }^{2}{x}^{2}+1}}{\sqrt {\arccos \left ( \mu \,x \right ) }}}+ \left ( \left ( -m-2 \right ) \LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,{\it \_b} \right ) \right ) +\arccos \left ( \mu \,{\it \_b} \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,{\it \_b} \right ) \right ) - \left ( \arccos \left ( \mu \,{\it \_b} \right ) \right ) ^{m+3/2} \right ) b\sqrt {-{{\it \_b}}^{2}{\mu }^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -\sqrt {\arccos \left ( \mu \,x \right ) }\sqrt {\arccos \left ( \mu \,{\it \_b} \right ) }bx\LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) +a\sqrt {\arccos \left ( \mu \,{\it \_b} \right ) }y+\arccos \left ( \mu \,{\it \_b} \right ) b{\it \_b}\,\LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,{\it \_b} \right ) \right ) \right ) \right ) } \right ) \right ) ^{k}\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {b \left ( \left ( m+2 \right ) \LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) \sqrt {-{\mu }^{2}{x}^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -bx\LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) \arccos \left ( \mu \,x \right ) +ya\sqrt {\arccos \left ( \mu \,x \right ) } \right ) }{a\mu \, \left ( m+2 \right ) \sqrt {\arccos \left ( \mu \,x \right ) }}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a} \left ( \arccos \left ( {\frac {\nu }{ \left ( m+2 \right ) a\mu } \left ( {\frac {b \left ( \left ( m+2 \right ) \LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) \sqrt {-{\mu }^{2}{x}^{2}+1}}{\sqrt {\arccos \left ( \mu \,x \right ) }}}-b \left ( {\frac { \left ( m+2 \right ) \LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) }{\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }}}-\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }\LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) + \left ( \arccos \left ( \mu \,{\it \_a} \right ) \right ) ^{m+1} \right ) \sqrt {-{{\it \_a}}^{2}{\mu }^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -\sqrt {\arccos \left ( \mu \,x \right ) }bx\LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,x \right ) \right ) +ya+\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }b{\it \_a}\,\LommelS 1 \left ( 1/2+m,1/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) \right ) \right ) } \right ) \right ) ^{k}}{d{\it \_a}}}}\]
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