42.1.8 problem 3.24 (c)

Internal problem ID [6830]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.24 (c)
Date solved : Monday, January 27, 2025 at 02:31:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 44

Order:=6; 
dsolve(diff(y(x),x$2)+(exp(x)-1)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}-\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 49

AsymptoticDSolveValue[D[y[x],{x,2}]+(Exp[x]-1)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (-\frac {x^5}{40}-\frac {x^4}{12}+x\right )+c_1 \left (-\frac {x^5}{120}-\frac {x^4}{24}-\frac {x^3}{6}+1\right ) \]