Question

Solve the differential equation (i.e find the equation for f(x)).
1.)f'(s)=10s-12s^3,f(3)=2
2.)f"(x)=sinx,f'(0)=1,f(0)=6

Answer 1

\[\begin{align*}\frac{df(s)}{ds}&=10s-12s^3\\
\int df(s)&=\int(10s-12s^3)~ds\\
f(s)&=\frac{10s^2}{2}-\frac{12s^4}{4}+C\\
f(s)&=5s^2-3s^4+C
\end{align*}\]
Given that \(f(3)=2\), you have
\[2=5(3)^2-3(3)^4+C\]
Solve for \(C\).

Answer 2

\[\begin{align*}
f''(x)&=\sin x\\
\int f''(x)~dx&=\int\sin x~dx\\
f'(x)&=-\cos x+C_1\\
\int f'(x)~dx&=-\int(\cos x-C_1)~dx\\
f(x)&=-\sin x+C_1x+C_2
\end{align*}\]
Given that \(f'(0)=1\) and \(f(0)=6\), you have
\[\begin{cases}
1=-\cos 0+C_1\\
6=-\sin 0+C_1(0)+C_2
\end{cases}\]
Solve for \(C_1\) and \(C_2\).