67.24.9 problem 34.5 (i)

Internal problem ID [16976]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.5 (i)
Date solved : Thursday, October 02, 2025 at 01:41:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 54

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

\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{18} x^{4}+\frac {3}{160} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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

\[ y(x)\to c_2 \left (\frac {x^5}{60}+\frac {x^4}{24}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {3 x^5}{160}+\frac {x^4}{18}+\frac {x^3}{12}+\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.260 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = C_{2} x + C_{1} + O\left (x^{6}\right ) \]