ANSWER
The particular solution is:
[tex]y=2-2 \cos(x)[/tex]
EXPLANATION
The given Ordinary Differential Equation is
[tex]y'=2 \sin(x)[/tex]
The general solution to this Differential equation is:
[tex]y=C-2 \cos(x)[/tex]
To find the particular solution, we need to apply the initial conditions (ICs)
[tex]y( \frac{\pi}{3} ) = 1[/tex]
This implies that;
[tex]C-2 \cos( \frac{\pi}{3} ) = 1[/tex]
[tex]C-2( \frac{1}{2} )= 1[/tex]
[tex]C-1= 1[/tex]
[tex]C= 1 + 1 = 2[/tex]
Hence the particular solution is
[tex]y=2-2 \cos(x)[/tex]
