61.34.9 problem 9

Internal problem ID [12773]
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 : 9
Date solved : Tuesday, January 28, 2025 at 04:19:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 42

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

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