85.73.3 problem 1 (c)

Internal problem ID [22956]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. B Exercises at page 316
Problem number : 1 (c)
Date solved : Thursday, October 02, 2025 at 09:16:56 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }&=y+{\mathrm e}^{x} \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

Order:=6; 
ode:=diff(y(x),x) = exp(x)+y(x); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 29

ode=D[y[x],{x,1}]==Exp[x]+y[x]; 
ic={y[0]==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {x^5}{24}+\frac {x^4}{6}+\frac {x^3}{2}+x^2+x \]

✓ Sympy. Time used: 0.193 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = x + x^{2} + \frac {x^{3}}{2} + \frac {x^{4}}{6} + \frac {x^{5}}{24} + O\left (x^{6}\right ) \]