Internal problem ID [7144]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-x y^{\prime }-y x=x} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 54
dsolve(diff(y(x),x$2)-x*diff(y(x),x)-x*y(x)-x=0,y(x), singsol=all)
\[ y \left (x \right ) = \pi \,{\mathrm e}^{-2-x} c_{1} \left (x +2\right ) \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right )-i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{\frac {x \left (x +2\right )}{2}} c_{1} -1+{\mathrm e}^{-x} \left (x +2\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.689 (sec). Leaf size: 216
DSolve[y''[x]-x*y'[x]-x*y[x]-x==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} e^{-\frac {1}{2} (x+2)^2} \left (2 \sqrt {2} e^{\frac {x^2}{2}+x+2} (x+2) \int _1^x\left (\frac {e^{K[1]} K[1]}{\sqrt {2}}-\frac {1}{2} e^{-\frac {1}{2} K[1]^2-K[1]-2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {(K[1]+2)^2}}{\sqrt {2}}\right ) K[1] \sqrt {(K[1]+2)^2}\right )dK[1]-\sqrt {2 \pi } \sqrt {(x+2)^2} \left (c_2 e^{\frac {x^2}{2}+x+2}+x+1\right ) \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )+2 e^{\frac {x^2}{2}+x+2} \left (e^x (x+1)+\sqrt {2} c_1 (x+2)+c_2 e^{\frac {1}{2} (x+2)^2}\right )\right ) \]