# Thread: Use integral calculus to solve differential equation problems

1. ## Use integral calculus to solve differential equation problems

Find the equation of the curve that satisfies the differential equation f"(x) = 6x - 12 and which passes through the points (0, -8) and (1, -1).

2. Originally Posted by LilDragonfly

Find the equation of the curve that satisfies the differential equation f"(x) = 6x - 12 and which passes through the points (0, -8) and (1, -1).
$\displaystyle f(x)=x^3-6x^2+ax+b$
Sooooo:

$\displaystyle f(0)=-8 = 0^3-6*0^2+a*0+b=b$
Thus b = -8.

$\displaystyle f(1)=-1=(1)^3-6(1)^2+a(1)-8$
gives:
$\displaystyle -1=1-6+a-8=-13-a$
or
$\displaystyle a = 12$

-Dan

3. I do not know if you understood what topsquak said, but when you have,
$\displaystyle f''(x)=6x-12$
It means if you integrate both sides you have,
$\displaystyle f'(x)=3x^2-12x+a$ with some constant $\displaystyle a$,
the reason to why f''(x) becomes f'(x) is because when you integrate a derivative of a function you get back the function (definition of integral). So when you have f''(x) integrating once would yield f'(x) so you need one more time:
$\displaystyle f(x)=x^3-6x^2+ax+b$
Then use topsqaurk post to complete the problem.