Internal problem ID [7287]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 63.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {\frac {x y^{\prime \prime }}{1-x}+y=\cos \left (x \right )} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 169
dsolve(x/(1-x)*diff(y(x),x$2)+y(x)=cos(x),y(x), singsol=all)
\[ y \left (x \right ) = -\left (\left (\operatorname {BesselI}\left (0, -x \right )+\operatorname {BesselI}\left (1, -x \right )\right ) \left (\int -\frac {\cos \left (x \right ) \left (\operatorname {BesselK}\left (0, -x \right )-\operatorname {BesselK}\left (1, -x \right )\right ) \left (x -1\right )}{x \left (\operatorname {BesselI}\left (0, x\right ) \left (x +1\right ) \operatorname {BesselK}\left (1, -x \right )+1-\left (x +1\right ) \operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right )\right )}d x \right )+\left (-\operatorname {BesselK}\left (0, -x \right )+\operatorname {BesselK}\left (1, -x \right )\right ) \left (\int -\frac {\cos \left (x \right ) \left (\operatorname {BesselI}\left (0, x\right )-\operatorname {BesselI}\left (1, x\right )\right ) \left (x -1\right )}{x \left (\operatorname {BesselI}\left (0, x\right ) \left (x +1\right ) \operatorname {BesselK}\left (1, -x \right )+1-\left (x +1\right ) \operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right )\right )}d x \right )+\operatorname {BesselK}\left (1, -x \right ) c_{1} -\operatorname {BesselK}\left (0, -x \right ) c_{1} -\operatorname {BesselI}\left (0, -x \right ) c_{2} -\operatorname {BesselI}\left (1, -x \right ) c_{2} \right ) x \]
✓ Solution by Mathematica
Time used: 8.805 (sec). Leaf size: 133
DSolve[x/(1-x)*y''[x]+y[x]==Cos[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} x \left (\operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right ) \int _1^x2 \sqrt {\pi } (\operatorname {BesselI}(0,K[1])-\operatorname {BesselI}(1,K[1])) \cos (K[1]) (K[1]-1)dK[1]+e^x (\operatorname {BesselI}(0,x)-\operatorname {BesselI}(1,x)) \int _1^x-2 e^{-K[2]} \sqrt {\pi } \cos (K[2]) \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 K[2]\right ) (K[2]-1)dK[2]+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right )+c_2 e^x \operatorname {BesselI}(0,x)-c_2 e^x \operatorname {BesselI}(1,x)\right ) \]