56.2.9 problem 9

Internal problem ID [8813]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 9
Date solved : Monday, January 27, 2025 at 05:03:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 56

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

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

✓ Solution by Mathematica

Time used: 3.077 (sec). Leaf size: 226

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

\[ y(x)\to \frac {1}{2} e^{-\frac {1}{2} (x+2)^2} \left (2 \sqrt {2} e^{\frac {x^2}{2}+x+2} (x+2) \int _1^x\left (\frac {e^{K[1]} K[1]^2}{\sqrt {2}}-\frac {1}{2} e^{-\frac {1}{2} K[1]^2-K[1]-2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {(K[1]+2)^2}}{\sqrt {2}}\right ) K[1]^2 \sqrt {(K[1]+2)^2}\right )dK[1]-\sqrt {2 \pi } \sqrt {(x+2)^2} \left (x^2+c_2 e^{\frac {x^2}{2}+x+2}+x+1\right ) \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )+2 e^{\frac {x^2}{2}+x+2} \left (e^x \left (x^2+x+1\right )+\sqrt {2} c_1 (x+2)+c_2 e^{\frac {1}{2} (x+2)^2}\right )\right ) \]