Internal problem ID [6392]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-x y^{\prime }-y x -x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 58
dsolve(diff(y(x),x$2)-x*diff(y(x),x)-x*y(x)-x^2=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-x} \left (x +2\right ) c_{2}+\left (\left (x +2\right ) \pi \erf \left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right ) {\mathrm e}^{-x -2}-i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{\frac {x \left (x +2\right )}{2}}\right ) c_{1}+1-x \]
✓ Solution by Mathematica
Time used: 1.541 (sec). Leaf size: 140
DSolve[y''[x]-x*y'[x]-x*y[x]-x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} \left (\sqrt {2} (x+2) \int _1^x\frac {1}{2} e^{K[1]} K[1]^2 \left (\sqrt {2}-2 F\left (\frac {K[1]+2}{\sqrt {2}}\right ) (K[1]+2)\right )dK[1]+e^x \left (x^2+x+1\right )+\sqrt {2} c_1 (x+2)+c_2 e^{\frac {1}{2} (x+2)^2}\right )-\sqrt {2} (x+2) \left (x^2+c_2 e^{\frac {x^2}{2}+x+2}+x+1\right ) F\left (\frac {x+2}{\sqrt {2}}\right ) \\ \end{align*}