Answer:
[tex]y=2-2\cos(x)[/tex]
Step-by-step explanation:
We have the differential:
[tex]y^\prime=2\sin(x)[/tex]
With the general solution:
[tex]y=C-2\cos(x)[/tex]
And we want to find the particular solution such that it satisfies the initial condition:
[tex]\displaystyle y\Big(\frac{\pi}{3}\Big)=1[/tex]
So, we have:
[tex]y=C-2\cos(x)[/tex]
Substituting π/3 for x and 1 for y yields:
[tex]\displaystyle 1=C-2\cos\Big( \frac{\pi}{3} \Big)[/tex]
Solve for C. Evaluate:
[tex]\displaystyle 1=C-2(\frac{1}{2})[/tex]
Simplify:
[tex]1=C-1[/tex]
Hence:
[tex]C=2[/tex]
Therefore, our particular solution will be:
[tex]y=2-2\cos(x)[/tex]
Hence, our answer is C
