45.1.3 problem 2. Using series method

Internal problem ID [9500]
Book : A course in Ordinary Differential Equations. by Stephen A. Wirkus, Randall J. Swift. CRC Press NY. 2015. 2nd Edition
Section : Chapter 8. Series Methods. section 8.2. The Power Series Method. Problems Page 603
Problem number : 2. Using series method
Date solved : Tuesday, September 30, 2025 at 06:19:43 PM
CAS classification : [[_linear, `class A`]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 24

Order:=8; 
ode:=diff(y(x),x)-2*y(x) = x^2; 
ic:=[y(1) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=1);
 

\[ y = 1+3 \left (x -1\right )+4 \left (x -1\right )^{2}+3 \left (x -1\right )^{3}+\frac {3}{2} \left (x -1\right )^{4}+\frac {3}{5} \left (x -1\right )^{5}+\frac {1}{5} \left (x -1\right )^{6}+\frac {2}{35} \left (x -1\right )^{7}+\operatorname {O}\left (\left (x -1\right )^{8}\right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 60

ode=D[y[x],x]-2*y[x]==x^2; 
ic={y[1]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,7}]
 

\[ y(x)\to \frac {2}{35} (x-1)^7+\frac {1}{5} (x-1)^6+\frac {3}{5} (x-1)^5+\frac {3}{2} (x-1)^4+3 (x-1)^3+4 (x-1)^2+3 (x-1)+1 \]

✓ Sympy. Time used: 0.195 (sec). Leaf size: 56

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

\[ y{\left (x \right )} = -2 + 4 \left (x - 1\right )^{2} + 3 \left (x - 1\right )^{3} + \frac {3 \left (x - 1\right )^{4}}{2} + \frac {3 \left (x - 1\right )^{5}}{5} + \frac {\left (x - 1\right )^{6}}{5} + \frac {2 \left (x - 1\right )^{7}}{35} + 3 x + O\left (x^{8}\right ) \]