Internal problem ID [11117]
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: 19.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a -\lambda \right ) y^{\prime }+y b \,{\mathrm e}^{2 \lambda x}=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 53
dsolve(diff(y(x),x$2)+(a*exp(lambda*x)-lambda)*diff(y(x),x)+b*exp(2*lambda*x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {\left (a -\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\lambda x}}{2 \lambda }}+c_{2} {\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\lambda x}}{2 \lambda }} \]
✓ Solution by Mathematica
Time used: 0.133 (sec). Leaf size: 61
DSolve[y''[x]+(a*Exp[\[Lambda]*x]-\[Lambda])*y'[x]+b*Exp[2*\[Lambda]*x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {\left (\sqrt {a^2-4 b}+a\right ) e^{\lambda x}}{2 \lambda }} \left (c_2 e^{\frac {\sqrt {a^2-4 b} e^{\lambda x}}{\lambda }}+c_1\right ) \]