56.2.4 problem 4

Internal problem ID [8808]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 4
Date solved : Monday, January 27, 2025 at 05:03:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x y^{\prime }-y x -x^{2}-x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 55

dsolve(diff(y(x),x$2)-x*diff(y(x),x)-x*y(x)-x^2-x=0,y(x), singsol=all)
 

\[ y = -\pi \,{\mathrm e}^{-x -2} c_{1} \left (x +2\right ) \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right )+i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{\frac {x \left (x +2\right )}{2}} c_{1} +{\mathrm e}^{-x} \left (x +2\right ) c_{2} -x \]

✓ Solution by Mathematica

Time used: 1.759 (sec). Leaf size: 84

DSolve[D[y[x],{x,2}]-x*D[y[x],x]-x*y[x]-x^2-x==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} e^{-x} \left (-\sqrt {2 \pi } c_2 \sqrt {(x+2)^2} \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )-2 e^x x+2 \sqrt {2} c_1 (x+2)+2 c_2 e^{\frac {1}{2} (x+2)^2}\right ) \]