61.28.12 problem 72

Internal problem ID [12572]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 72
Date solved : Tuesday, January 28, 2025 at 03:21:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 21

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

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

✓ Solution by Mathematica

Time used: 0.145 (sec). Leaf size: 70

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

\[ y(x)\to \frac {e^{-a x} x^{\frac {1}{2} \left (-b-\sqrt {(b-1)^2}+1\right )} \left (c_2 x^{\sqrt {(b-1)^2}}+\sqrt {(b-1)^2} c_1\right )}{\sqrt {(b-1)^2}} \]