7.15.2 problem 2

Internal problem ID [458]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 2
Date solved : Saturday, March 29, 2025 at 04:54:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+x^{2} y^{\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: 0.006 (sec). Leaf size: 54

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

\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{12} x^{3}+\frac {1}{9} x^{4}+\frac {13}{480} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{24} x^{4}+\frac {7}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

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

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

\[ y(x)\to c_2 \left (\frac {7 x^5}{120}-\frac {x^4}{24}-\frac {x^3}{3}+x\right )+c_1 \left (\frac {13 x^5}{480}+\frac {x^4}{9}-\frac {x^3}{12}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.901 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{40} - \frac {x^{2}}{6} + 1\right ) + C_{1} + O\left (x^{6}\right ) \]