Internal problem ID [6426]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 43.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-y^{\prime } x -y x -x=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 55
dsolve(diff(y(x),x$2)-x*diff(y(x),x)-x*y(x)-x=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-x} \left (x +2\right ) c_{2} +\left (\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right ) \pi \left (x +2\right ) {\mathrm e}^{-x -2}-i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{\frac {x \left (x +2\right )}{2}}\right ) c_{1} -1 \]
✓ Solution by Mathematica
Time used: 0.231 (sec). Leaf size: 104
DSolve[y''[x]-x*y'[x]-x*y[x]-x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {2} e^{-x} (x+2) \left (\int _1^x\frac {1}{2} e^{K[1]} K[1] \left (\sqrt {2}-2 \operatorname {DawsonF}\left (\frac {K[1]+2}{\sqrt {2}}\right ) (K[1]+2)\right )dK[1]+c_1\right )-\left (c_2 e^{\frac {x^2}{2}+x+2}+x+1\right ) \left (\sqrt {2} (x+2) \operatorname {DawsonF}\left (\frac {x+2}{\sqrt {2}}\right )-1\right ) \\ \end{align*}