Second Order differential equation involving chain rule

[Woolyabyss]

Homework Statement

Solve (d^2x)/(dt^2) = 2x(9 + x^2) given that dx/dt = 9 when x = 0 and x = 3 when t = 0

Homework Equations

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

 

[vela]

Homework Statement

Solve (d^2x)/(dt^2) = 2x(9 + x^2) given that dx/dt = 9 when x = 0 and x = 3 when t = 0

Homework Equations

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 it's no longer a part of our course.

Any help would be appreciated.

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

 

[Woolyabyss]

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

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)