61.27.20 problem 30

Internal problem ID [12530]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 30
Date solved : Tuesday, January 28, 2025 at 03:19:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 55

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

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

✓ Solution by Mathematica

Time used: 0.154 (sec). Leaf size: 86

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

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