61.26.3 problem 3

Internal problem ID [12503]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2 Equations Containing Power Functions. page 213
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 03:18:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 22

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

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

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 43

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

\[ y(x)\to c_1 \operatorname {ParabolicCylinderD}\left (-1,\sqrt {2} \sqrt {a} x\right )+c_2 \operatorname {ParabolicCylinderD}\left (0,i \sqrt {2} \sqrt {a} x\right ) \]