61.34.23 problem 23

Internal problem ID [12787]
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 : 23
Date solved : Tuesday, January 28, 2025 at 04:19:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Solution by Maple

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

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 0.824 (sec). Leaf size: 164

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

\[ y(x)\to (-1)^{-\frac {\lambda }{\mu }} 2^{\frac {\lambda +\mu }{2 \mu }} \left (\left (e^x\right )^{\lambda }\right )^{\frac {\lambda -1}{2 \lambda }} \left (e^x\right )^{\frac {1}{2}-\frac {\mu }{2}} e^{-\frac {a \left (e^x\right )^{\lambda }}{\lambda }} \left (\left (e^x\right )^{\mu }\right )^{\frac {\lambda +\mu }{2 \mu }} \left (-\frac {c \left (e^x\right )^{\mu }}{\mu ^2}\right )^{-\frac {\lambda }{2 \mu }} \left (c_1 (-1)^{\lambda /\mu } \operatorname {BesselI}\left (\frac {\lambda }{\mu },2 \sqrt {-\frac {c \left (e^x\right )^{\mu }}{\mu ^2}}\right )+c_2 K_{\frac {\lambda }{\mu }}\left (2 \sqrt {-\frac {c \left (e^x\right )^{\mu }}{\mu ^2}}\right )\right ) \]