61.29.26 problem 135

Internal problem ID [12635]
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 : 135
Date solved : Tuesday, January 28, 2025 at 08:02:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.185 (sec). Leaf size: 79

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

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

✓ Solution by Mathematica

Time used: 0.242 (sec). Leaf size: 172

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

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