61.29.22 problem 131

Internal problem ID [12631]
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 : 131
Date solved : Tuesday, January 28, 2025 at 08:02:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.208 (sec). Leaf size: 71

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

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

✓ Solution by Mathematica

Time used: 0.461 (sec). Leaf size: 169

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

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i b}{\sqrt {a}}+\sqrt {(\lambda -1)^2-4 c}+1\right ),\sqrt {(\lambda -1)^2-4 c}+1,2 i \sqrt {a} x\right )+c_2 L_{\frac {1}{2} \left (-\frac {i b}{\sqrt {a}}-\sqrt {(\lambda -1)^2-4 c}-1\right )}^{\sqrt {(\lambda -1)^2-4 c}}\left (2 i \sqrt {a} x\right )\right ) \exp \left (\int _1^x\frac {-\lambda -2 i \sqrt {a} K[1]+\sqrt {(\lambda -1)^2-4 c}+1}{2 K[1]}dK[1]\right ) \]