Internal problem ID [11111]
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: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+2 a \,{\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left ({\mathrm e}^{\lambda x} a +\lambda \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 32
dsolve(diff(y(x),x$2)+2*a*exp(lambda*x)*diff(y(x),x)+a*exp(lambda*x)*(a*exp(lambda*x)+lambda)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}+c_{2} {\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} x \]
✓ Solution by Mathematica
Time used: 0.109 (sec). Leaf size: 26
DSolve[y''[x]+2*a*Exp[\[Lambda]*x]*y'[x]+a*Exp[\[Lambda]*x]*(a*Exp[\[Lambda]*x]+\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (c_2 x+c_1) e^{-\frac {a e^{\lambda x}}{\lambda }} \]