61.34.8 problem 8

Internal problem ID [12772]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.3-1. Equations with exponential functions
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 04:18:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{2 a x} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.453 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.087 (sec). Leaf size: 78

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

\[ y(x)\to \frac {\sqrt {a} e^{-\frac {a x}{2}} \left (2 c_1 \cos \left (\frac {\sqrt {b e^{2 a x}}}{a}\right )+c_2 \sin \left (\frac {\sqrt {b e^{2 a x}}}{a}\right )\right )}{\sqrt {2} \sqrt [4]{b e^{2 a x}}} \]