Differential equation Show that the differential equation $$\frac{dy}{dx} =\frac{e^x+x}{\sin y+2}$$ has a solution satisfying $y(0) = \pi$. To do this, separate variables and integrate to get an equation implicitly relating $x$ and $y$.

$$
\frac {dy}{dx} = \frac {e^x + x}{\sin y + 2} \\
(\sin y + 2) dy = (e^x + x) dx \\
\int (\sin y + 2) dy = \int (e^x + x) dx \\
-\cos y + 2y = e^x + \frac {x^2}2 + C
$$
Now, substitute $y(0) = \pi$
$$
1 + 2\pi = 1 + C
$$
from which you can find that $C = 2\pi$. 
Final answer is
$$
2y - \cos y = e^x + \frac {x^2}2 + 2\pi
$$
is a solution of a given ODE and satisfies $y(0) = \pi$.
