61.34.25 problem 25

Internal problem ID [12789]
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
Date solved : Tuesday, January 28, 2025 at 08:24:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left (c \,{\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+b^{2}-b \lambda \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.429 (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)
 

\[ y = c_{1} {\mathrm e}^{\frac {-2 b \lambda x +{\mathrm e}^{\lambda x} \sqrt {a^{2}-4 c}-{\mathrm e}^{\lambda x} a}{2 \lambda }}+c_{2} {\mathrm e}^{-\frac {2 b \lambda x +{\mathrm e}^{\lambda x} \sqrt {a^{2}-4 c}+{\mathrm e}^{\lambda x} a}{2 \lambda }} \]

✓ Solution by Mathematica

Time used: 1.138 (sec). Leaf size: 102

DSolve[D[y[x],{x,2}]+(a*Exp[\[Lambda]*x]+2*b-\[Lambda])*D[y[x],x]+(c*Exp[2*\[Lambda]*x]+a*b*Exp[\[Lambda]*x]+b^2-b*\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\left (e^{\lambda x}\right )^{-\frac {b}{\lambda }} e^{-\frac {\left (\sqrt {a^2-4 c}+a\right ) e^{\lambda x}+2 b}{2 \lambda }} \left (c_2 \lambda e^{\frac {\sqrt {a^2-4 c} e^{\lambda x}}{\lambda }}+c_1 \sqrt {a^2-4 c}\right )}{\sqrt {a^2-4 c}} \]