59.3.8 problem David Saunders 1981 paper. Example 1

Internal problem ID [10010]
Book : Collection of Kovacic problems
Section : section 3. Problems from Kovacic related papers
Problem number : David Saunders 1981 paper. Example 1
Date solved : Monday, January 27, 2025 at 06:18:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 46

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

\[ y(x)\to e^{-\frac {x^2}{2}} \left (-\sqrt {\pi } c_2 e^{x^2} x \text {erf}(x)+c_1 e^{x^2} x-c_2\right ) \]