Internal problem ID [6393]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-x y^{\prime }-y x -x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 252
dsolve(diff(y(x),x$2)-x*diff(y(x),x)-x*y(x)-x^3=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-x} \left (x +2\right ) c_{2}+\left (\left (x +2\right ) \pi \erf \left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right ) {\mathrm e}^{-x -2}-i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{\frac {x \left (x +2\right )}{2}}\right ) c_{1}+\frac {{\mathrm e}^{-\frac {1}{2} x^{2}-3 x -2} \left (i {\mathrm e}^{\frac {\left (x +2\right )^{2}}{2}} \sqrt {2}\, \left (x +2\right ) \left (\int -{\mathrm e}^{-\frac {x \left (x +2\right )}{2}} \left (-\pi \left (x +2\right ) {\mathrm e}^{-2} \erf \left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right )+i {\mathrm e}^{\frac {x \left (4+x \right )}{2}} \sqrt {2}\, \sqrt {\pi }\right ) x^{3}d x \right )-2 i \erf \left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right ) \left (x +2\right ) \pi ^{\frac {3}{2}} \erf \left (\frac {\sqrt {2}\, \left (x +1\right )}{2}\right ) {\mathrm e}^{\frac {1}{2}+\frac {1}{2} x^{2}+2 x}+i \sqrt {2}\, \left (x^{2}+x +2\right ) \left (x +2\right ) \pi \,{\mathrm e}^{x} x \erf \left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right )-2 \sqrt {2}\, \erf \left (\frac {\sqrt {2}\, \left (x +1\right )}{2}\right ) \pi \,{\mathrm e}^{x^{2}+4 x +\frac {5}{2}}+2 \sqrt {\pi }\, x \,{\mathrm e}^{\frac {1}{2} x^{2}+3 x +2} \left (x^{2}+x +2\right )\right )}{2 \sqrt {\pi }} \]
✓ Solution by Mathematica
Time used: 1.925 (sec). Leaf size: 237
DSolve[y''[x]-x*y'[x]-x*y[x]-x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-\frac {1}{2} (x+2)^2} \left (\sqrt {2} (x+2) \left (2 e^{\frac {x^2}{2}+x+2} \int _1^x\frac {1}{2} e^{K[1]} K[1]^3 \left (\sqrt {2}-2 F\left (\frac {K[1]+2}{\sqrt {2}}\right ) (K[1]+2)\right )dK[1]-\sqrt {\pi } x \left (x^2+x+2\right ) \text {Erfi}\left (\frac {x+2}{\sqrt {2}}\right )\right )-2 e^{(x+1) (x+2)} \left (\sqrt {e \pi } \left (\sqrt {2}-2 (x+2) F\left (\frac {x+2}{\sqrt {2}}\right )\right ) \text {Erf}\left (\frac {x+1}{\sqrt {2}}\right )+\sqrt {2} e^2 c_2 (x+2) F\left (\frac {x+2}{\sqrt {2}}\right )\right )+2 e^{\frac {x^2}{2}+x+2} \left (e^x x \left (x^2+x+2\right )+\sqrt {2} c_1 (x+2)+c_2 e^{\frac {1}{2} (x+2)^2}\right )\right ) \\ \end{align*}