2. $[\mathbf{3}]$ Let $f$ be a function such that $f(0)=1, f^{\prime}(0)=2$, and
$$
f^{\prime \prime}(t)=4 f^{\prime}(t)-3 f(t)+1
$$
for all $t$. Compute the 4th derivative of $f$, evaluated at 0 .
Answer: 54 Putting $t=0$ gives $f^{\prime \prime}(0)=6$. By differentiating both sides, we get $f^{(3)}(t)=4 f^{\prime \prime}(t)-$ $3 f^{\prime}(t)$ and $f^{(3)}(0)=4 \cdot 6-3 \cdot 2=18$. Similarly, $f^{(4)}(t)=4 f^{(3)}(t)-3 f^{\prime \prime}(t)$ and $f^{(4)}(0)=4 \cdot 18-3 \cdot 6=54$.