# Second Order differential equation involving chain rule

1. Oct 25, 2013

### Woolyabyss

1. The problem statement, all variables and given/known data
Solve (d^2x)/(dt^2) = 2x(9 + x^2) given that dx/dt = 9 when x = 0 and x = 3 when t = 0

2. Relevant equations

3. The attempt at a solution

v = dx/dt ....................... dv/dx = d^2x/dt^2

dv/dx = v(dv/dx)

v(dv/dx) = 18x +2x^3

integrating and evaluating using limits and then you get

(v^2/)2 - 81/2 = 9x^2 +.5x^4

multiply by 2 and add 81 to both sides

v^2 = 18x^2 + x^4 + 81

dx/dt = v = + or - (18^2 + x^4 +81)^(1/2)

this where I have a problem generally when we did these questions I would be able to just get the root without needing to use the square root symbol. I'm not sure how to integrate polynomials with a power.Does this require integration by substitution? Because I know its no longer apart of our course.

Any help would be appreciated

2. Oct 25, 2013

### vela

Staff Emeritus
$x^4 + 18 x^2 + 81$ is a perfect square.

3. Oct 25, 2013

### Woolyabyss

Oh right i didn't see that. So it would be ( x^2 + 9 )^2

and then when you get the square root (x^2 + 9)

Last edited: Oct 25, 2013