# Differential Systems

Calculus Level 5

$\large 2\frac {dx}{dt} + 2x + \frac{dy}{dt} - y = 3t$

$\large \frac{dx}{dt} + x + \frac{dy}{dt} + y = 1$

$$x(0) = 1$$; $$y(0) = 3$$

Given that $$x$$ and $$y$$ are both functions of $$t$$, the value of $$\frac{dy}{dx}$$ when $$x= 1 + \frac{3}{e} - \frac{2}{e^3}$$ can be expressed in the form $$\large -\frac{ae^{b} + c}{ae^{b} - be^{d} + c}$$ where $$a$$, $$b$$, $$c$$, and $$d$$ are distinct positive integers. Determine $$abcd -(a+b+c+d)$$.

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