78.17.2 problem 1 (b)

Internal problem ID [18337]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 27. Series Solutions of First Order Equations. Problems at page 208
Problem number : 1 (b)
Date solved : Monday, March 31, 2025 at 05:25:52 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }+y&=1 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 58

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 69

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

\[ y(x)\to \frac {x^5}{120}-\frac {x^4}{24}+\frac {x^3}{6}-\frac {x^2}{2}+c_1 \left (-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right )+x \]

✓ Sympy. Time used: 0.650 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = - x \left (C_{1} - 1\right ) + \frac {x^{2} \left (C_{1} - 1\right )}{2} - \frac {x^{3} \left (C_{1} - 1\right )}{6} + \frac {x^{4} \left (C_{1} - 1\right )}{24} - \frac {x^{5} \left (C_{1} - 1\right )}{120} + C_{1} + O\left (x^{6}\right ) \]