61.34.24 problem 24

Internal problem ID [12788]
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 : 24
Date solved : Tuesday, January 28, 2025 at 04:19:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 54

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

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

✓ Solution by Mathematica

Time used: 0.335 (sec). Leaf size: 82

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

\[ y(x)\to \frac {\left (c_2 e^{x \sqrt {b^2-4 c}}+c_1 \sqrt {b^2-4 c}\right ) e^{-\frac {a e^{\lambda x}}{\lambda }-\frac {1}{2} x \left (\sqrt {b^2-4 c}+b\right )}}{\sqrt {b^2-4 c}} \]