59.1.496 problem 512

Internal problem ID [9668]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 512
Date solved : Monday, January 27, 2025 at 06:12:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 x y^{\prime }+\left (2 x^{2}+4\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 48

dsolve(diff(y(x),x$2)+3*x*diff(y(x),x)+(4+2*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.201 (sec). Leaf size: 50

DSolve[D[y[x],{x,2}]+3*x*D[y[x],x]+(4+2*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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