Let us solve the given differential equation step-by-step to find the correct formula for the function \( y = f(t) \).
We are given:
[tex]\[ y' = t^4 \][/tex]
[tex]\[ y(0) = 3 \][/tex]
First, we need to find the general solution to the differential equation by integrating \( y' \).
[tex]\[
y = \int t^4 \, dt
\][/tex]
Using the power rule of integration, we can integrate \( t^4 \):
[tex]\[
y = \frac{t^5}{5} + C
\][/tex]
Where \( C \) is the constant of integration.
Next, we use the initial condition \( y(0) = 3 \) to find the value of \( C \):
[tex]\[
y(0) = \frac{0^5}{5} + C = 3
\][/tex]
[tex]\[
C = 3
\][/tex]
Thus, the particular solution to the differential equation is:
[tex]\[
y = \frac{t^5}{5} + 3
\][/tex]
Now we check if any of the given options match this solution:
1. \(\frac{t^3}{3}\)
2. \(\frac{t^3}{5} + 3\)
3. \(t^3 - 3\)
4. \(t^5 + 3\)
5. \(t^3 + 3\)
Our particular solution \( \frac{t^5}{5} + 3 \) does not match any of the given options.
Thus, the correct result is none of the given formulas match the derived function [tex]\( y = \frac{t^5}{5} + 3 \)[/tex].
