61.29.8 problem 117

Internal problem ID [12617]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 117
Date solved : Tuesday, January 28, 2025 at 08:02:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 38

dsolve(x^2*diff(y(x),x$2)-(a^2*x^4+a*(2*b-1)*x^2+b*(b+1))*y(x)=0,y(x), singsol=all)
 

\[ y = x^{-b} {\mathrm e}^{-\frac {a \,x^{2}}{2}} \left (c_{2} \Gamma \left (b +\frac {1}{2}\right )-c_{2} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.259 (sec). Leaf size: 70

DSolve[x^2*D[y[x],{x,2}]-(a^2*x^4+a*(2*b-1)*x^2+b*(b+1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} x^{-b} e^{-\frac {a x^2}{2}-b} \left (a c_2 x^{2 b+3} \left (-a x^2\right )^{-b-\frac {3}{2}} \Gamma \left (b+\frac {1}{2},-a x^2\right )+2 c_1\right ) \]