61.34.14 problem 14

Internal problem ID [12778]
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 : 14
Date solved : Tuesday, January 28, 2025 at 08:24:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.371 (sec). Leaf size: 36

dsolve(diff(y(x),x$2)+(a+b)*exp(lambda*x)*diff(y(x),x)+a*exp(lambda*x)*(b*exp(lambda*x)+lambda)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.227 (sec). Leaf size: 44

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

\[ y(x)\to e^{-\frac {a e^{\lambda x}}{\lambda }-1} \left (c_2 \operatorname {ExpIntegralEi}\left (\frac {(a-b) e^{x \lambda }}{\lambda }\right )+e c_1\right ) \]