# Use integral calculus to solve differential equation problems

• May 17th 2006, 01:33 AM
LilDragonfly
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).
• May 17th 2006, 04:27 AM
topsquark
Quote:

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
• May 17th 2006, 05:33 AM
ThePerfectHacker
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.