78.17.5 problem 4

Internal problem ID [18332]
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 : 4
Date solved : Thursday, March 13, 2025 at 11:54:23 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=1+y^{2} \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: 0.002 (sec). Leaf size: 14

Order:=6; 
ode:=diff(y(x),x) = 1+y(x)^2; 
ic:=y(0) = 0; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = x +\frac {1}{3} x^{3}+\frac {2}{15} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 19

ode=D[y[x],x]==1+y[x]^2; 
ic={y[0]==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {2 x^5}{15}+\frac {x^3}{3}+x \]

✓ Sympy. Time used: 0.660 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**2 + Derivative(y(x), x) - 1,0) 
ics = {y(0): 0} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)
 

\[ y{\left (x \right )} = x + \frac {x^{3}}{3} + \frac {2 x^{5}}{15} + O\left (x^{6}\right ) \]