61.34.17 problem 17

Internal problem ID [12781]
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 : 17
Date solved : Tuesday, January 28, 2025 at 04:19:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 35

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

\[ y(x)\to \frac {e^{\frac {b e^{a x}}{a}} \left (b c_2 e^{a x}+a c_1\right )}{a} \]