59.1.498 problem 514

Internal problem ID [9670]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 514
Date solved : Monday, January 27, 2025 at 06:12:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 62

dsolve(diff(y(x),x$2)-3*x*diff(y(x),x)+(5+2*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \left (x^{6}-15 x^{4}+45 x^{2}-15\right ) \left (c_{1} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+c_{2} \right ) {\mathrm e}^{\frac {x^{2}}{2}}-2 \,{\mathrm e}^{x^{2}} c_{1} x \left (x^{2}-11\right ) \left (x^{2}-3\right ) \]

✓ Solution by Mathematica

Time used: 0.267 (sec). Leaf size: 74

DSolve[D[y[x],{x,2}]-3*x*D[y[x],x]+(5+2*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{\frac {x^2}{2}} \left (x^6-15 x^4+45 x^2-15\right ) \left (c_2 \int _1^x\frac {e^{\frac {K[1]^2}{2}}}{\left (K[1]^6-15 K[1]^4+45 K[1]^2-15\right )^2}dK[1]+c_1\right ) \]