61.34.31 problem 31

Internal problem ID [12795]
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 : 31
Date solved : Tuesday, January 28, 2025 at 08:24:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 81

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

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

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved