Question

solve the initial - value problem $\frac{dy}{dx}=cos(x),y(\frac{pi}{3}) = 5.$

Explanation:

Step1: Integrate both sides

Integrate $\frac{dy}{dx}=\cos(x)$ with respect to $x$. We know that $\int dy=\int\cos(x)dx$. So $y = \sin(x)+C$.

Step2: Use the initial - condition

Substitute $x = \frac{\pi}{3}$ and $y = 5$ into $y=\sin(x)+C$. Since $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$, we have $5=\frac{\sqrt{3}}{2}+C$.

Step3: Solve for C

$C = 5-\frac{\sqrt{3}}{2}$.

Step4: Write the final solution

The solution of the initial - value problem is $y=\sin(x)+5 - \frac{\sqrt{3}}{2}$.

Answer:

$y=\sin(x)+5-\frac{\sqrt{3}}{2}$