# Math Help - Solve the differential equation?

1. ## Solve the differential equation?

Solve the differential equation?

f"(x)=x^2 , f'(0)=6, f(0)=3

2. Consider the rule

$\int x^n ~dx = \frac{x^{n+1}}{n+1}+C$

so

$f'(x) = \int x^2 ~dx = \frac{x^3}{3}+C$

now you must solve for C using $f'(0)=6$

$f'(x) = \frac{x^3}{3}+C$

$6 = \frac{0^3}{3}+C$

now

$C=6$ and $f'(x) = \frac{x^3}{3}+6$

Now repeat this step again for $f(x) =\int \frac{x^3}{3}+6~dx$ using $f(0)=3$