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 : Sunday, March 30, 2025 at 11:23:56 AM
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*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 44

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

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

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

ode=D[y[x],{x,2}]+(Exp[x]-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 1.092 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((exp(x) - 1)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4} \left (1 - e^{x}\right )^{2}}{24} - \frac {x^{2} e^{x}}{2} + \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (- \frac {x^{2} e^{x}}{6} + \frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]