Internal problem ID [11123]
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: 25.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{x \lambda }+2 b -\lambda \right ) y^{\prime }+\left ({\mathrm e}^{2 x \lambda } c +a b \,{\mathrm e}^{x \lambda }+b^{2}-b \lambda \right ) y=0} \]
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 74
dsolve(diff(y(x),x$2)+(a*exp(lambda*x)+2*b-lambda)*diff(y(x),x)+(c*exp(2*lambda*x)+a*b*exp(lambda*x)+b^2-b*lambda)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {2 \lambda x b -{\mathrm e}^{\lambda x} \sqrt {a^{2}-4 c}+a \,{\mathrm e}^{\lambda x}}{2 \lambda }}+c_{2} {\mathrm e}^{-\frac {2 \lambda x b +{\mathrm e}^{\lambda x} \sqrt {a^{2}-4 c}+a \,{\mathrm e}^{\lambda x}}{2 \lambda }} \]
✓ Solution by Mathematica
Time used: 2.119 (sec). Leaf size: 97
DSolve[y''[x]+(a*Exp[\[Lambda]*x]+2*b-\[Lambda])*y'[x]+(c*Exp[2*\[Lambda]*x]+a*b*Exp[\[Lambda]*x]+b^2-b*\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\left (e^{\lambda x}\right )^{-\frac {b}{\lambda }} e^{-\frac {\left (\sqrt {a^2-4 c}+a\right ) e^{\lambda x}}{2 \lambda }} \left (c_2 \lambda e^{\frac {\sqrt {a^2-4 c} e^{\lambda x}}{\lambda }}+c_1 \sqrt {a^2-4 c}\right )}{\sqrt {a^2-4 c}} \]