68.16.19 problem 21

Internal problem ID [17812]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 21
Date solved : Thursday, October 02, 2025 at 02:28:28 PM
CAS classification : [[_2nd_order, _missing_x], _Van_der_Pol]

\begin{align*} y^{\prime \prime }+\left (-1+y^{2}\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 16

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

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

✗ Mathematica

ode=D[y[x],{x,2}]+(y[x]^2-1)*D[y[x],x]+y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

Not solved

✗ Sympy

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

ValueError : ODE (y(x)**2 - 1)*Derivative(y(x), x) + y(x) + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular