61.29.5 problem 114

Internal problem ID [12614]
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 : 114
Date solved : Tuesday, January 28, 2025 at 08:02:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 29

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

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

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 38

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

\[ y(x)\to c_1 M_{-b,b-\frac {1}{2}}(2 a x)+c_2 W_{-b,b-\frac {1}{2}}(2 a x) \]