Internal problem ID [6534]
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]]
Solve \begin {gather*} \boxed {\frac {x y^{\prime \prime }}{1-x}+y-\cos \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.468 (sec). Leaf size: 154
dsolve(x/(1-x)*diff(y(x),x$2)+y(x)=cos(x),y(x), singsol=all)
\[ y \relax (x ) = \left (\BesselI \left (0, -x \right )+\BesselI \left (1, -x \right )\right ) x c_{2}+x \left (\BesselK \left (0, -x \right )-\BesselK \left (1, -x \right )\right ) c_{1}-\left (\left (\BesselI \left (0, -x \right )+\BesselI \left (1, -x \right )\right ) \left (\int \frac {\left (\BesselK \left (0, -x \right )-\BesselK \left (1, -x \right )\right ) \cos \relax (x ) \left (x -1\right )}{x \left (-\BesselI \left (0, x\right ) \BesselK \left (1, -x \right )+\BesselK \left (0, -x \right ) \BesselI \left (1, x\right )\right )}d x \right )-\left (\int \frac {\left (\BesselI \left (0, x\right )-\BesselI \left (1, x\right )\right ) \cos \relax (x ) \left (x -1\right )}{x \left (-\BesselI \left (0, x\right ) \BesselK \left (1, -x \right )+\BesselK \left (0, -x \right ) \BesselI \left (1, x\right )\right )}d x \right ) \left (\BesselK \left (0, -x \right )-\BesselK \left (1, -x \right )\right )\right ) x \]
✓ Solution by Mathematica
Time used: 5.095 (sec). Leaf size: 95
DSolve[x/(1-x)*y''[x]+y[x]==Cos[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} x \text {HypergeometricU}\left (\frac {1}{2},2,2 x\right ) \left (\int _1^x2 \sqrt {\pi } (I_0(K[1])-I_1(K[1])) \cos (K[1]) (K[1]-1)dK[1]+c_1\right )+x (I_0(x)-I_1(x)) \left (\int _1^x-((K_0(K[2])+K_1(K[2])) \cos (K[2]) (K[2]-1))dK[2]+c_2\right ) \\ \end{align*}