78.17.6 problem 5

Internal problem ID [18341]
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 : 5
Date solved : Monday, March 31, 2025 at 05:25:56 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }&=x -y \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: 0.003 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 32

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

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

✓ Sympy. Time used: 0.634 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + y(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)
 

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