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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