61.29.28 problem 137

Internal problem ID [12637]
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 : 137
Date solved : Tuesday, January 28, 2025 at 03:23:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 48

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

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

✓ Solution by Mathematica

Time used: 0.096 (sec). Leaf size: 48

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

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