61.34.4 problem 4

Internal problem ID [12768]
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 : 4
Date solved : Tuesday, January 28, 2025 at 08:24:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.916 (sec). Leaf size: 76

dsolve(diff(y(x),x$2)-(a^2*exp(2*x)+a*(2*b+1)*exp(x)+b^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.038 (sec). Leaf size: 59

DSolve[D[y[x],{x,2}]-(a^2*Exp[2*x]+a*(2*b+1)*Exp[x]+b^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{a e^x} \left (e^x\right )^b \left (c_2 \int _1^{e^x}\exp \left (\int _1^{K[2]}-\frac {2 b+2 a K[1]+1}{K[1]}dK[1]\right )dK[2]+c_1\right ) \]