59.1.313 problem 318

Internal problem ID [9485]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 318
Date solved : Monday, January 27, 2025 at 06:03:20 PM
CAS classification : [_Hermite]

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 42

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

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

✓ Solution by Mathematica

Time used: 0.203 (sec). Leaf size: 43

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

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