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I have this system of differential equations: $$z=y'$$ $$y=-z'-4$$

How would Cauchy's problem look for this equation, if I have z(0)=-4 and y(0)=1 ?

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    $\begingroup$ What exactly do you ask for; for the solution $(y(t), z(t))$? What have you tried? $\endgroup$ Jun 15, 2014 at 10:53

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Assuming that $y$ and $z$ are each functions of $t$, we've got:

$(1) \ \ z(t)=y'(t)$

$(2)\ \ y(t)=-\color{green}{z'(t)}-4$.

Differentiate $(1)$ wrt. $t$, to give:

$(3) \ \ \color{green}{z'(t)=y''(t)}$.

Now, sub $(3)$ into $(2)$ to give:

$(4) \ \ y(t)=-\color{green}{y''(t)}-4$.

Re-arrange $(4)$ to give:

$(*)\ \ y''(t)+y(t)=4$

Now solve $(*)$ (it's an inhomogeneous second order ODE)!

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