44.17.11 problem 1(f) solving using series

Internal problem ID [9368]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.2. Series Solutions of First-Order Differential Equations Page 162
Problem number : 1(f) solving using series
Date solved : Tuesday, September 30, 2025 at 06:17:51 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-y&=x^{2} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 70

Order:=8; 
ode:=diff(y(x),x)-y(x) = x^2; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) y \left (0\right )+\frac {x^{3}}{3}+\frac {x^{4}}{12}+\frac {x^{5}}{60}+\frac {x^{6}}{360}+\frac {x^{7}}{2520}+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 87

ode=D[y[x],x]-y[x]==x^2; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {x^7}{2520}+\frac {x^6}{360}+\frac {x^5}{60}+\frac {x^4}{12}+\frac {x^3}{3}+c_1 \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right ) \]

✓ Sympy. Time used: 0.193 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 - y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=8)
 

\[ y{\left (x \right )} = \frac {x^{3} \left (C_{1} + 2\right )}{6} + \frac {x^{4} \left (C_{1} + 2\right )}{24} + \frac {x^{5} \left (C_{1} + 2\right )}{120} + \frac {x^{6} \left (C_{1} + 2\right )}{720} + \frac {x^{7} \left (C_{1} + 2\right )}{5040} + C_{1} + C_{1} x + \frac {C_{1} x^{2}}{2} + O\left (x^{8}\right ) \]