Particular Solution of System of Constant Coefficient Linear 1st Order ODEs

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Theorem

Consider the system of linear first order ordinary differential equations with constant coefficients:

\(\text {(1)}: \quad\) \(\ds \dfrac {\d y} {\d x} + a y + b z\) \(=\) \(\ds 0\)
\(\text {(2)}: \quad\) \(\ds \dfrac {\d x} {\d z} + c y + d z\) \(=\) \(\ds 0\)

Let $(1)$ and $(2)$ have the following $n$ initial conditions:

$(3): \quad y = y_0, z = z_0$

when $x = x_0$.


Then there exists exactly one particular solution of $(1)$ and $(2)$ which satisfies $(3)$.


Proof



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