I am looking at the motion of particles in force fields (so acceleration). With a given vector acceleration
$$\vec{a}(x,y)= \begin{bmatrix} P(x,y) \\ Q(x,y) \end{bmatrix} \textrm{with initial velocity } \begin{bmatrix} v_{x_0} \\ v_{y_0} \end{bmatrix} \textrm{and position} \begin{bmatrix} x_0 \\ y_0 \end{bmatrix}, $$
I have the system of second order differential equations
$$ x^{\prime\prime}(t)=P(x(t),y(t)) $$ $$ y^{\prime\prime}(t)=Q(x(t),y(t)). $$
This isn't too hard to solve for simple acceleration components, but how would I approach, for example, a system such as
$$ x^{\prime\prime}(t)=\sin(y(t)) $$ $$ y^{\prime\prime}(t)=-2x(t)+y(t)\textrm{ }? $$