Solve the differential equation?
f"(x)=x^2 , f'(0)=6, f(0)=3
Please help
Consider the rule
$\displaystyle \int x^n ~dx = \frac{x^{n+1}}{n+1}+C$
so
$\displaystyle f'(x) = \int x^2 ~dx = \frac{x^3}{3}+C$
now you must solve for C using $\displaystyle f'(0)=6$
$\displaystyle f'(x) = \frac{x^3}{3}+C$
$\displaystyle 6 = \frac{0^3}{3}+C$
now
$\displaystyle C=6$ and $\displaystyle f'(x) = \frac{x^3}{3}+6$
Now repeat this step again for $\displaystyle f(x) =\int \frac{x^3}{3}+6~dx $ using $\displaystyle f(0)=3$