67.24.21 problem 34.8 b(i)

Internal problem ID [16988]
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 : 34.8 b(i)
Date solved : Thursday, October 02, 2025 at 01:41:33 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }-y \,{\mathrm e}^{x}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 53

Order:=10; 
ode:=diff(y(x),x)-exp(x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}+\frac {203}{720} x^{6}+\frac {877}{5040} x^{7}+\frac {23}{224} x^{8}+\frac {1007}{17280} x^{9}\right ) y \left (0\right )+O\left (x^{10}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 61

ode=D[y[x],x]-Exp[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,9}]
 

\[ y(x)\to c_1 \left (\frac {1007 x^9}{17280}+\frac {23 x^8}{224}+\frac {877 x^7}{5040}+\frac {203 x^6}{720}+\frac {13 x^5}{30}+\frac {5 x^4}{8}+\frac {5 x^3}{6}+x^2+x+1\right ) \]

✓ Sympy. Time used: 0.349 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)*exp(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=10)
 

\[ y{\left (x \right )} = C_{1} + C_{1} x + C_{1} x^{2} + \frac {5 C_{1} x^{3}}{6} + \frac {5 C_{1} x^{4}}{8} + \frac {13 C_{1} x^{5}}{30} + \frac {203 C_{1} x^{6}}{720} + \frac {877 C_{1} x^{7}}{5040} + \frac {23 C_{1} x^{8}}{224} + \frac {1007 C_{1} x^{9}}{17280} + O\left (x^{10}\right ) \]