61.34.18 problem 18

Internal problem ID [12782]
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 : 18
Date solved : Tuesday, January 28, 2025 at 08:24:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.628 (sec). Leaf size: 79

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

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

✓ Solution by Mathematica

Time used: 0.121 (sec). Leaf size: 129

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

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