61.29.6 problem 115

Internal problem ID [12615]
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 : 115
Date solved : Tuesday, January 28, 2025 at 08:02:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.182 (sec). Leaf size: 53

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

\[ y = c_{1} \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i \sqrt {a}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i \sqrt {a}\, x \right ) \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 88

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

\[ y(x)\to c_1 M_{-\frac {i b}{2 \sqrt {a}},-\frac {1}{2} i \sqrt {4 c-1}}\left (2 i \sqrt {a} x\right )+c_2 W_{-\frac {i b}{2 \sqrt {a}},-\frac {1}{2} i \sqrt {4 c-1}}\left (2 i \sqrt {a} x\right ) \]