74.16.19 problem 21

Internal problem ID [16525]
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 : Tuesday, January 28, 2025 at 09:10:12 AM
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*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 16

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

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

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 26

AsymptoticDSolveValue[{D[y[x],{x,2}]+(y[x]^2-1)*D[y[x],x]+y[x]==0,{y[0]==0,Derivative[1][y][0] ==1}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to -\frac {x^5}{8}-\frac {x^4}{8}+\frac {x^2}{2}+x \]