56.2.44 problem 43

Internal problem ID [8848]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 43
Date solved : Monday, January 27, 2025 at 05:06:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 52

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

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

✓ Solution by Mathematica

Time used: 0.701 (sec). Leaf size: 216

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

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