45.1.1 problem 1. Using series method

Internal problem ID [9498]
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 : 1. Using series method
Date solved : Tuesday, September 30, 2025 at 06:19:40 PM
CAS classification : [[_Riccati, _special]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 24

Order:=8; 
ode:=diff(y(x),x) = y(x)^2-x; 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 1+x +\frac {1}{2} x^{2}+\frac {2}{3} x^{3}+\frac {7}{12} x^{4}+\frac {11}{20} x^{5}+\frac {22}{45} x^{6}+\frac {559}{1260} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Mathematica. Time used: 8.172 (sec). Leaf size: 1113

ode=D[y[x],x]==y[x]^2-x; 
ic={y[0]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

Too large to display

✓ Sympy. Time used: 0.200 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = 1 + x + \frac {x^{2}}{2} + \frac {2 x^{3}}{3} + \frac {7 x^{4}}{12} + \frac {11 x^{5}}{20} + \frac {22 x^{6}}{45} + \frac {559 x^{7}}{1260} + O\left (x^{8}\right ) \]