6.5.19 7.1

6.5.19.1 [1324] Problem 1

problem number 1324

Added April 13, 2019.

Problem Chapter 5.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = w + c_1 \arcsin ^k(\lambda x) + c_2 \arcsin ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcSin[lambda*x]^k+c2*ArcSin[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \sin ^{-1}(\lambda K[1])^k+\text {c2} \sin ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) =  w(x,y)+c1*arcsin(lambda*x)^k+c2*arcsin(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {{\it \_a}}{a}}}} \left ( {\it c2}\, \left ( \arcsin \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \beta }{a}} \right ) \right ) ^{n}+{\it c1}\, \left ( \arcsin \left ( {\it \_a}\,\lambda \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]

____________________________________________________________________________________

6.5.19.2 [1325] Problem 2

problem number 1325

Added April 13, 2019.

Problem Chapter 5.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = c w + \arcsin ^k(\lambda x) \arcsin ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ ArcSin[lambda*x]^k*ArcSin[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) =  c*w(x,y)+ arcsin(lambda*x)^k*arcsin(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \arcsin \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \arcsin \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \beta }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]

____________________________________________________________________________________

6.5.19.3 [1326] Problem 3

problem number 1326

Added April 13, 2019.

Problem Chapter 5.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = \left ( c_1 \arcsin (\lambda _1 x) + c_2 \arcsin (\lambda _2 y)\right ) w+ s_1 \arcsin ^n(\beta _1 x)+ s_2 \arcsin ^k(\beta _2 y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcSin[lambda1*x] + c2*ArcSin[lambda2*y])*w[x,y]+ s1*ArcSin[beta1*x]^n+ s2*ArcSin[beta2*y]^k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\frac {\text {c1} \sqrt {1-\text {lambda1}^2 x^2}}{a \text {lambda1}}+\frac {\text {c1} x \sin ^{-1}(\text {lambda1} x)}{a}+\frac {\text {c2} \sqrt {1-\text {lambda2}^2 y^2}}{b \text {lambda2}}+\frac {\text {c2} y \sin ^{-1}(\text {lambda2} y)}{b}\right ) \left (\int _1^x\frac {\exp \left (-\frac {b \text {c1} \text {lambda2} \sin ^{-1}(\text {lambda1} K[1]) K[1] \text {lambda1}+\text {c2} \text {lambda2} \sin ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) (a y+b (K[1]-x)) \text {lambda1}+a \text {c2} \sqrt {-y^2 \text {lambda2}^2-\frac {b^2 (x-K[1])^2 \text {lambda2}^2}{a^2}+\frac {2 b y (x-K[1]) \text {lambda2}^2}{a}+1} \text {lambda1}+b \text {c1} \text {lambda2} \sqrt {1-\text {lambda1}^2 K[1]^2}}{a b \text {lambda1} \text {lambda2}}\right ) \left (\text {s2} \sin ^{-1}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )^k+\text {s1} \sin ^{-1}(\text {beta1} K[1])^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arcsin(lambda1*x) + c2*arcsin(lambda2*y))*w(x,y)+ s1*arcsin(beta1*x)^n+ s2*arcsin(beta2*y)^k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{{\frac {1}{a\lambda 1\,b\lambda 2} \left ( -\sqrt {-{\frac { \left ( \left ( \lambda 2\,y-1 \right ) a-\lambda 2\,b \left ( x-{\it \_a} \right ) \right ) \left ( \left ( \lambda 2\,y+1 \right ) a-\lambda 2\,b \left ( x-{\it \_a} \right ) \right ) }{{a}^{2}}}}a{\it c2}\,\lambda 1- \left ( \lambda 1\,{\it c2}\, \left ( \left ( {\it \_a}-x \right ) b+ya \right ) \arcsin \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \lambda 2}{a}} \right ) +{\it c1}\,b \left ( \arcsin \left ( \lambda 1\,{\it \_a} \right ) {\it \_a}\,\lambda 1+\sqrt {-{{\it \_a}}^{2}{\lambda 1}^{2}+1} \right ) \right ) \lambda 2 \right ) }}} \left ( {\it s2}\, \left ( \arcsin \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \beta 2}{a}} \right ) \right ) ^{k}+{\it s1}\, \left ( \arcsin \left ( \beta 1\,{\it \_a} \right ) \right ) ^{n} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {\sqrt {-{\lambda 2}^{2}{y}^{2}+1}a{\it c2}\,\lambda 1+\lambda 2\, \left ( b{\it c1}\,\sqrt {-{\lambda 1}^{2}{x}^{2}+1}+\lambda 1\, \left ( a\arcsin \left ( \lambda 2\,y \right ) y{\it c2}+bx{\it c1}\,\arcsin \left ( \lambda 1\,x \right ) \right ) \right ) }{a\lambda 1\,b\lambda 2}}}}\]

____________________________________________________________________________________

6.5.19.4 [1327] Problem 4

problem number 1327

Added April 13, 2019.

Problem Chapter 5.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \arcsin ^m(\mu x) w_y = c \arcsin ^k(\nu x) w + p \arcsin ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*ArcSin[mu*x]^m*D[w[x, y], y] == c*ArcSin[nu*x]^k*w[x,y]+p*ArcSin[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\frac {i c \sin ^{-1}(\nu x)^k \left (\sin ^{-1}(\nu x)^2\right )^{-k} \left (\left (-i \sin ^{-1}(\nu x)\right )^k \text {Gamma}\left (k+1,i \sin ^{-1}(\nu x)\right )-\left (i \sin ^{-1}(\nu x)\right )^k \text {Gamma}\left (k+1,-i \sin ^{-1}(\nu x)\right )\right )}{2 a \nu }\right ) \left (\int _1^x\frac {\exp \left (\frac {i c \sin ^{-1}(\nu K[1])^k \left (\sin ^{-1}(\nu K[1])^2\right )^{-k} \left (\left (i \sin ^{-1}(\nu K[1])\right )^k \text {Gamma}\left (k+1,-i \sin ^{-1}(\nu K[1])\right )-\left (-i \sin ^{-1}(\nu K[1])\right )^k \text {Gamma}\left (k+1,i \sin ^{-1}(\nu K[1])\right )\right )}{2 a \nu }\right ) p \sin ^{-1}\left (\frac {\beta \left (i b \sin ^{-1}(\mu x)^m \left (\left (i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,-i \sin ^{-1}(\mu x)\right )-\left (-i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,i \sin ^{-1}(\mu x)\right )\right ) \left (\sin ^{-1}(\mu x)^2\right )^{-m}+2 a \mu y-i b \sin ^{-1}(\mu K[1])^m \left (\sin ^{-1}(\mu K[1])^2\right )^{-m} \left (\left (i \sin ^{-1}(\mu K[1])\right )^m \text {Gamma}\left (m+1,-i \sin ^{-1}(\mu K[1])\right )-\left (-i \sin ^{-1}(\mu K[1])\right )^m \text {Gamma}\left (m+1,i \sin ^{-1}(\mu K[1])\right )\right )\right )}{2 a \mu }\right )^n}{a}dK[1]+c_1\left (y+\frac {i b \sin ^{-1}(\mu x)^m \left (\sin ^{-1}(\mu x)^2\right )^{-m} \left (\left (i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,-i \sin ^{-1}(\mu x)\right )-\left (-i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,i \sin ^{-1}(\mu x)\right )\right )}{2 a \mu }\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*arcsin(mu*x)^m*diff(w(x,y),y) = c*arcsin(nu*x)^k*w(x,y)+p*arcsin(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 ( -\arcsin \left ( {\frac {\beta \, \left ( \mu \,{\it \_f}+1 \right ) \left ( \mu \,{\it \_f}-1 \right ) }{\mu \, \left ( m+1 \right ) \left ( {{\it \_f}}^{2}{\mu }^{2}-1 \right ) a} \left ( -{2}^{-m}\arcsin \left ( \mu \,{\it \_f} \right ) b{2}^{m} \left ( -{\frac {\LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,{\it \_f} \right ) \right ) }{\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }}}+ \left ( \arcsin \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \right ) \sqrt {-{{\it \_f}}^{2}{\mu }^{2}+1}+\mu \, \left ( a \left ( m+1 \right ) \int \!{\frac {b \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x-{\frac {{2}^{-m}b{\it \_f}\,{2}^{m}\LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,{\it \_f} \right ) \right ) }{\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }}}-{2}^{m}{2}^{-m}b\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }\LommelS 1 \left ( 1/2+m,3/2,\arcsin \left ( \mu \,{\it \_f} \right ) \right ) m{\it \_f}-a \left ( m+1 \right ) y \right ) \right ) } \right ) \right ) ^{n}{{\rm e}^{{\frac {c{2}^{-k} \left ( \nu \,{\it \_f}-1 \right ) {2}^{k} \left ( \nu \,{\it \_f}+1 \right ) }{ \left ( k+1 \right ) a\nu \, \left ( {{\it \_f}}^{2}{\nu }^{2}-1 \right ) } \left ( \arcsin \left ( \nu \,{\it \_f} \right ) \left ( {\frac {\LommelS 1 \left ( 3/2+k,1/2,\arcsin \left ( \nu \,{\it \_f} \right ) \right ) }{\sqrt {\arcsin \left ( \nu \,{\it \_f} \right ) }}}- \left ( \arcsin \left ( \nu \,{\it \_f} \right ) \right ) ^{k} \right ) \sqrt {-{{\it \_f}}^{2}{\nu }^{2}+1}-{\it \_f}\,\nu \, \left ( \sqrt {\arcsin \left ( \nu \,{\it \_f} \right ) }k\LommelS 1 \left ( k+1/2,3/2,\arcsin \left ( \nu \,{\it \_f} \right ) \right ) +{\frac {\LommelS 1 \left ( 3/2+k,1/2,\arcsin \left ( \nu \,{\it \_f} \right ) \right ) }{\sqrt {\arcsin \left ( \nu \,{\it \_f} \right ) }}} \right ) \right ) }}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {-b \left ( -\arcsin \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,x \right ) \right ) + \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) \sqrt {-{\mu }^{2}{x}^{2}+1}+ \left ( -x\LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,x \right ) \right ) b-\LommelS 1 \left ( 1/2+m,3/2,\arcsin \left ( \mu \,x \right ) \right ) \arcsin \left ( \mu \,x \right ) mxb+a\sqrt {\arcsin \left ( \mu \,x \right ) }y \left ( m+1 \right ) \right ) \mu }{\sqrt {\arcsin \left ( \mu \,x \right ) }\mu \,a \left ( m+1 \right ) }} \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \arcsin \left ( \nu \,x \right ) \right ) ^{k}c}{a}}\,{\rm d}x}}\]

____________________________________________________________________________________

6.5.19.5 [1328] Problem 5

problem number 1328

Added April 13, 2019.

Problem Chapter 5.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \arcsin ^m(\mu x) w_y = c \arcsin ^k(\nu y) w + p \arcsin ^n(\beta x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*ArcSin[mu*x]^m*D[w[x, y], y] == c*ArcSin[nu*y]^k*w[x,y]+p*ArcSin[beta*x]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \sin ^{-1}\left (\frac {\nu \left (i b \sin ^{-1}(\mu x)^m \left (\left (i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,-i \sin ^{-1}(\mu x)\right )-\left (-i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,i \sin ^{-1}(\mu x)\right )\right ) \left (\sin ^{-1}(\mu x)^2\right )^{-m}+2 a \mu y-i b \sin ^{-1}(\mu K[1])^m \left (\sin ^{-1}(\mu K[1])^2\right )^{-m} \left (\left (i \sin ^{-1}(\mu K[1])\right )^m \text {Gamma}\left (m+1,-i \sin ^{-1}(\mu K[1])\right )-\left (-i \sin ^{-1}(\mu K[1])\right )^m \text {Gamma}\left (m+1,i \sin ^{-1}(\mu K[1])\right )\right )\right )}{2 a \mu }\right )^k}{a}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {c \sin ^{-1}\left (\frac {\nu \left (i b \sin ^{-1}(\mu x)^m \left (\left (i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,-i \sin ^{-1}(\mu x)\right )-\left (-i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,i \sin ^{-1}(\mu x)\right )\right ) \left (\sin ^{-1}(\mu x)^2\right )^{-m}+2 a \mu y-i b \sin ^{-1}(\mu K[1])^m \left (\sin ^{-1}(\mu K[1])^2\right )^{-m} \left (\left (i \sin ^{-1}(\mu K[1])\right )^m \text {Gamma}\left (m+1,-i \sin ^{-1}(\mu K[1])\right )-\left (-i \sin ^{-1}(\mu K[1])\right )^m \text {Gamma}\left (m+1,i \sin ^{-1}(\mu K[1])\right )\right )\right )}{2 a \mu }\right )^k}{a}dK[1]\right ) p \sin ^{-1}(\beta K[2])^n}{a}dK[2]+c_1\left (y+\frac {i b \sin ^{-1}(\mu x)^m \left (\sin ^{-1}(\mu x)^2\right )^{-m} \left (\left (i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,-i \sin ^{-1}(\mu x)\right )-\left (-i \sin ^{-1}(\mu x)\right )^m \text {Gamma}\left (m+1,i \sin ^{-1}(\mu x)\right )\right )}{2 a \mu }\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*arcsin(mu*x)^m*diff(w(x,y),y) = c*arcsin(nu*y)^k*w(x,y)+p*arcsin(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 ( \arcsin \left ( {\it \_f}\,\beta \right ) \right ) ^{n}}{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( -\arcsin \left ( {\frac {\nu \, \left ( \mu \,{\it \_f}+1 \right ) \left ( \mu \,{\it \_f}-1 \right ) }{\mu \, \left ( m+1 \right ) \left ( {{\it \_f}}^{2}{\mu }^{2}-1 \right ) a} \left ( -{2}^{-m}\arcsin \left ( \mu \,{\it \_f} \right ) b{2}^{m} \left ( -{\frac {\LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,{\it \_f} \right ) \right ) }{\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }}}+ \left ( \arcsin \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \right ) \sqrt {-{{\it \_f}}^{2}{\mu }^{2}+1}+\mu \, \left ( a \left ( m+1 \right ) \int \!{\frac {b \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x-{\frac {{2}^{-m}b{\it \_f}\,{2}^{m}\LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,{\it \_f} \right ) \right ) }{\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }}}-{2}^{m}{2}^{-m}b\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }\LommelS 1 \left ( 1/2+m,3/2,\arcsin \left ( \mu \,{\it \_f} \right ) \right ) m{\it \_f}-a \left ( m+1 \right ) y \right ) \right ) } \right ) \right ) ^{k}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {-b \left ( -\arcsin \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,x \right ) \right ) + \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) \sqrt {-{\mu }^{2}{x}^{2}+1}+ \left ( -x\LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,x \right ) \right ) b-\LommelS 1 \left ( 1/2+m,3/2,\arcsin \left ( \mu \,x \right ) \right ) \arcsin \left ( \mu \,x \right ) mxb+a\sqrt {\arcsin \left ( \mu \,x \right ) }y \left ( m+1 \right ) \right ) \mu }{\sqrt {\arcsin \left ( \mu \,x \right ) }\mu \,a \left ( m+1 \right ) }} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a} \left ( -\arcsin \left ( {\frac {\nu \, \left ( \mu \,{\it \_b}+1 \right ) \left ( \mu \,{\it \_b}-1 \right ) }{\mu \, \left ( m+1 \right ) \left ( {{\it \_b}}^{2}{\mu }^{2}-1 \right ) a} \left ( -{2}^{-m}\arcsin \left ( \mu \,{\it \_b} \right ) b{2}^{m} \left ( -{\frac {\LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,{\it \_b} \right ) \right ) }{\sqrt {\arcsin \left ( \mu \,{\it \_b} \right ) }}}+ \left ( \arcsin \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \right ) \sqrt {-{{\it \_b}}^{2}{\mu }^{2}+1}+\mu \, \left ( a \left ( m+1 \right ) \int \!{\frac {b \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x-{\frac {{2}^{-m}b{\it \_b}\,{2}^{m}\LommelS 1 \left ( m+3/2,1/2,\arcsin \left ( \mu \,{\it \_b} \right ) \right ) }{\sqrt {\arcsin \left ( \mu \,{\it \_b} \right ) }}}-{2}^{m}{2}^{-m}b\sqrt {\arcsin \left ( \mu \,{\it \_b} \right ) }\LommelS 1 \left ( 1/2+m,3/2,\arcsin \left ( \mu \,{\it \_b} \right ) \right ) m{\it \_b}-a \left ( m+1 \right ) y \right ) \right ) } \right ) \right ) ^{k}}{d{\it \_b}}}}\]

____________________________________________________________________________________