solve the initial value problem.\n$\frac{ds}{dt}=cos t - sin t, s(\frac{pi}{2}) = 4$

Explanation:

Step1: Integrate the right - hand side

Integrate $\cos t-\sin t$ with respect to $t$. We know that $\int\cos tdt=\sin t + C_1$ and $\int(-\sin t)dt=\cos t + C_2$. So, $s(t)=\int(\cos t - \sin t)dt=\sin t+\cos t + C$.

Step2: Use the initial condition

Given $s(\frac{\pi}{2}) = 4$. Substitute $t = \frac{\pi}{2}$ into $s(t)=\sin t+\cos t + C$. We have $\sin(\frac{\pi}{2})+\cos(\frac{\pi}{2})+C=4$. Since $\sin(\frac{\pi}{2}) = 1$ and $\cos(\frac{\pi}{2}) = 0$, then $1 + 0+C=4$, which gives $C = 3$.

Answer:

$s(t)=\sin t+\cos t + 3$