find the particular solution of the differential equation that satisfies the initial condition(s).
$f(s)=10s - 4s^{3},\ f(3)=5$
$f(s)=$

Question

Accepted Answer

# Explanation:
## Step1: Integrate $f^{\prime}(s)$ to find $f(s)$
We know that if $f^{\prime}(s)=10s - 4s^{3}$, then by the power rule of integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$):
$$
\begin{align*}
f(s)&=\int(10s-4s^{3})ds\
&=10\int sds-4\int s^{3}ds\
&=10\times\frac{s^{2}}{2}-4\times\frac{s^{4}}{4}+C\
&=5s^{2}-s^{4}+C
\end{align*}
$$
## Step2: Use the initial condition $f(3) = 5$ to find $C$
Substitute $s = 3$ and $f(3)=5$ into $f(s)=5s^{2}-s^{4}+C$:
$$
\begin{align*}
5&=5\times3^{2}-3^{4}+C\
5&=5\times9 - 81+C\
5&=45-81+C\
5&=- 36+C
\end{align*}
$$
Solve for $C$: $C=5 + 36=41$

# Answer:
$f(s)=5s^{2}-s^{4}+41$

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QUESTION IMAGE

Question

Explanation:

Step1: Integrate \(f^{\prime}(s)\) to find \(f(s)\)

Step2: Use the initial condition \(f(3) = 5\) to find \(C\)

Answer: