# Thread: Integrate a function which includes the integral of itself

1. ## Integrate a function which includes the integral of itself

I've got the formula for the acceleration of a vehicle. Basically it looks like this:
$\displaystyle a = c_1 - c_2*v^2 - c_3$
Where a = acceleration
v = velocity
and the other are constants.
I need to integrate this formula, so I can get hold of the velocity (v), but don't know how. Numerical integration is fine!
I've got Matlab and Wolfram Mathematica.

Is this enough information? Hope you understand my problem!

2. Originally Posted by Woppe
I've got the formula for the acceleration of a vehicle. Basically it looks like this:
$\displaystyle a = c_1 - c_2*v^2 - c_3$
Where a = acceleration
v = velocity
and the other are constants.
I need to integrate this formula, so I can get hold of the velocity (v), but don't know how. Numerical integration is fine!
I've got Matlab and Wolfram Mathematica.

Is this enough information? Hope you understand my problem!
start with the fact that $\displaystyle a = \frac{dv}{dt}$ ... you'll have to separate variables to get velocity as a function of time.

also ... is the equation $\displaystyle a = (c_1-c_2)(v^2-c_3)$ , or is it as you have it written?

3. The equation is as I have written it.

I separated the variables to:
$\displaystyle \frac {dv}{c_1-c_2*v^2-c_3} = dt$ Correct?

I need one more hint what to do next...

4. Originally Posted by Woppe
The equation is as I have written it.

I separated the variables to:
$\displaystyle \frac {dv}{c_1-c_2*v^2-c_3} = dt$ Correct?

I need one more hint what to do next...
let $\displaystyle k = c_1 - c_3$

$\displaystyle \frac{dv}{k - c_2v^2} = dt$

$\displaystyle \frac{dv}{(\sqrt{k} + \sqrt{c_2}v)(\sqrt{k} - \sqrt{c_2}v)} = dt$

partial fractions looks like the way to go from here. might be messy.