# How do you solve the differential equation given f'(s)=6s-8s^3, f(2)=3?

Nov 11, 2016

$f \left(s\right) = 3 {s}^{2} - 2 {s}^{4} + 23$

#### Explanation:

Let $y = f \left(s\right) \implies \frac{\mathrm{dy}}{\mathrm{ds}} = 6 s - 8 {s}^{3}$

This is a first order separable DE, so we can separate the variables as follows:

$\int \mathrm{dy} = \int 6 s - 8 {s}^{3} \mathrm{ds}$

Integrating gives:

$y = 3 {s}^{2} - 2 {s}^{4} + C$

We know $f \left(2\right) = 3 \implies 3 = \left(3\right) \left(4\right) - \left(2\right) \left(16\right) + C$
$\therefore 3 = 12 - 32 + C \implies C = 23$

Hence, the solution is:

$y = 3 {s}^{2} - 2 {s}^{4} + 23$