System of Linear Differential Equations 1

Calculus Level 5

\[\dot { x } = 0.1 x+ 0.2y \\ \dot { y } = 0.3x+0.2y \] Given the solution to the system of ODEs above is in the form \[\left[ \begin{matrix} x \\ y \end{matrix} \right] =A\left[ \begin{matrix} a \\ -b \end{matrix} \right] { e }^{ -0.1t }+B\left[ \begin{matrix} c \\ d \end{matrix} \right] { e }^{ 0.4t }\] where \(a,b,c,d\) are all positive integers such that \(a,b \) are coprime, and \( c, d \) are coprime.

Calculate \(-ad-bc\).

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