Internal problem ID [6532]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 61.
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-\frac {1}{1-x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 144
dsolve(x/(1-x)*diff(y(x),x$2)+y(x)=1/(1-x),y(x), singsol=all)
\[ y \relax (x ) = \left (\BesselI \left (0, -x \right )+\BesselI \left (1, -x \right )\right ) x c_{2}+x \left (\BesselK \left (0, -x \right )-\BesselK \left (1, -x \right )\right ) c_{1}+\left (\left (-\BesselK \left (0, -x \right )+\BesselK \left (1, -x \right )\right ) \left (\int \frac {\BesselI \left (0, -x \right )+\BesselI \left (1, -x \right )}{x \left (-\BesselI \left (0, x\right ) \BesselK \left (1, -x \right )+\BesselK \left (0, -x \right ) \BesselI \left (1, x\right )\right )}d x \right )+\left (\int \frac {\BesselK \left (0, -x \right )-\BesselK \left (1, -x \right )}{x \left (-\BesselI \left (0, x\right ) \BesselK \left (1, -x \right )+\BesselK \left (0, -x \right ) \BesselI \left (1, x\right )\right )}d x \right ) \left (\BesselI \left (0, -x \right )+\BesselI \left (1, -x \right )\right )\right ) x \]
✓ Solution by Mathematica
Time used: 0.183 (sec). Leaf size: 89
DSolve[x/(1-x)*y''[x]+y[x]==1/(1-x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x (I_0(x)-I_1(x)) \left (\int _1^x(K_0(K[1])+K_1(K[1]))dK[1]+c_2\right )+e^{-x} x \text {HypergeometricU}\left (\frac {1}{2},2,2 x\right ) \left (-2 \sqrt {\pi } x \, _1F_2\left (\frac {1}{2};1,\frac {3}{2};\frac {x^2}{4}\right )+2 \sqrt {\pi } I_0(x)+c_1\right ) \\ \end{align*}