79.7.3 problem (b.2)

Internal problem ID [21104]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 2. Theory of First order differential equations. Excercises IV at page 89
Problem number : (b.2)
Date solved : Thursday, October 02, 2025 at 07:08:23 PM
CAS classification : [_quadrature]

\begin{align*} u^{\prime }&=u^{3} \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
ode:=diff(u(x),x) = u(x)^3; 
ic:=[u(0) = 1]; 
dsolve([ode,op(ic)],u(x),type='series',x=0);
 

\[ u = 1+x +\frac {3}{2} x^{2}+\frac {5}{2} x^{3}+\frac {35}{8} x^{4}+\frac {63}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 34

ode=D[u[x],x]==u[x]^3; 
ic={u[0]==1}; 
AsymptoticDSolveValue[{ode,ic},u[x],{x,0,5}]
 

\[ u(x)\to \frac {63 x^5}{8}+\frac {35 x^4}{8}+\frac {5 x^3}{2}+\frac {3 x^2}{2}+x+1 \]

✓ Sympy. Time used: 0.138 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
u = Function("u") 
ode = Eq(-u(x)**3 + Derivative(u(x), x),0) 
ics = {u(0): 1} 
dsolve(ode,func=u(x),ics=ics,hint="1st_power_series",x0=0,n=6)
 

\[ u{\left (x \right )} = 1 + x + \frac {3 x^{2}}{2} + \frac {5 x^{3}}{2} + \frac {35 x^{4}}{8} + \frac {63 x^{5}}{8} + O\left (x^{6}\right ) \]