Internal problem ID [11106]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{2 a x} y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 43
dsolve(diff(y(x),x$2)+a*diff(y(x),x)+b*exp(2*a*x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-a x} \sin \left (\frac {\sqrt {b}\, {\mathrm e}^{a x}}{a}\right )+c_{2} {\mathrm e}^{-a x} \cos \left (\frac {\sqrt {b}\, {\mathrm e}^{a x}}{a}\right ) \]
✓ Solution by Mathematica
Time used: 0.133 (sec). Leaf size: 78
DSolve[y''[x]+a*y'[x]+b*Exp[2*a*x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\sqrt {a} e^{-\frac {a x}{2}} \left (2 c_1 \cos \left (\frac {\sqrt {b e^{2 a x}}}{a}\right )+c_2 \sin \left (\frac {\sqrt {b e^{2 a x}}}{a}\right )\right )}{\sqrt {2} \sqrt [4]{b e^{2 a x}}} \]