Internal problem ID [5078]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\frac {y}{4}+\frac {x^{2}}{2}-\frac {1}{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 37
dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+1/4*y(x)=1/2*(1-x^2),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{2}}{\sqrt {x +\sqrt {x^{2}-1}}}+\sqrt {x +\sqrt {x^{2}-1}}\, c_{1}+\frac {2 x^{2}}{15}+\frac {14}{15} \]
✓ Solution by Mathematica
Time used: 7.617 (sec). Leaf size: 171
DSolve[(1-x^2)*y''[x]-x*y'[x]+1/4*y[x]==1/2*(1-x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to i \sinh \left (\frac {\sqrt {1-x^2} \text {ArcSin}(x)}{2 \sqrt {x^2-1}}\right ) \left (\int _1^x-i \cosh \left (\frac {\text {ArcSin}(K[2]) \sqrt {1-K[2]^2}}{2 \sqrt {K[2]^2-1}}\right ) \sqrt {K[2]^2-1}dK[2]+c_2\right )+\cosh \left (\frac {\sqrt {1-x^2} \text {ArcSin}(x)}{2 \sqrt {x^2-1}}\right ) \left (\int _1^x-\sqrt {K[1]^2-1} \sinh \left (\frac {\text {ArcSin}(K[1]) \sqrt {1-K[1]^2}}{2 \sqrt {K[1]^2-1}}\right )dK[1]+c_1\right ) \\ \end{align*}