61.34.19 problem 19

Internal problem ID [12783]
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
Date solved : Tuesday, January 28, 2025 at 04:19:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.469 (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)
 

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

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 61

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

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