Rewrite the 2nd oder non linear D.E as a series of 1st order equations

[beetle2]

Homework Statement

Rewrite the 2nd oder non linear D.E $\frac{d^2x}{dt^2}+x^2+x=0$
as a series of 1st order equations

Homework Equations

$a\frac{d^2x}{dt^2}+b\frac{dx}{dt}+cx=0$

$\frac{dx}{dt}=y$

$\frac{dy}{dt}=-\frac{c}{a}x-\frac{b}{a}y$

The Attempt at a Solution

a=1, b=0 , c=1

SO

$\frac{dx}{dt}=y$

$\frac{dy}{dt}=-\frac{1}{1}x-\frac{0}{1}y$ = $-x$

Is that right?

 

[hunt_mat]

You have two linear equations there, where is the non-linear term?

 

[HallsofIvy]

Homework Statement

Rewrite the 2nd oder non linear D.E $\frac{d^2x}{dt^2}+x^2+x=0$
as a series of 1st order equations

Homework Equations

$a\frac{d^2x}{dt^2}+b\frac{dx}{dt}+cx=0$

This is a linear equation and so not a good format to use for your problem.

$\frac{dx}{dt}=y$

$\frac{dy}{dt}=-\frac{c}{a}x-\frac{b}{a}y$

The Attempt at a Solution

a=1, b=0 , c=1

SO

$\frac{dx}{dt}=y$

$\frac{dy}{dt}=-\frac{1}{1}x-\frac{0}{1}y$ = $-x$

Is that right?

Don't just try to match up with memorized formulas (especially when the formula doesn't fit the problem).

You have defined y to be dx/dt so you know that $dy/dt= d^2x/dt^2$. Replace $d^2x/dt^2$ in $\frac{d^2x}{dt^2}+x^2+x=0$ with $dy/dx$ to get
$$\frac{dy}{dt}+ x^2+ x= 0$$
or
$$\frac{dy}{dt}= -x^2- x$$.