Solve $\frac{dx}{dt}=1$, $\frac{dy}{dt}=\cos(x(t))$

Question

Question: Solve $$\frac{dx}{d t}=1$$$$\frac{dy}{dt}=\cos(x(t))$$ Where $x(0)=x_0$ and $y(0)=y_0$

Answer: I have gotten $x(t)=t+x_0$, cant seem to get $y(t)$ should be a simple problem, but just cant do it.

Answer 1

You have $$y^\prime(t) = cos(t+x_0)$$ Thus: $$\begin{align} y(t) & = y_0 + \int_0^t y^\prime(\tilde t) d\tilde t \\ & = y_0 + \int_0^t \cos(\tilde t+x_0) d\tilde t \\ & = y_0 + [\sin(\tilde t + x_0)]_0^t \\ & = y_0 + \sin(t+x_0)-\sin(x_0)\end{align}$$