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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Sources
- 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 3$. Equations of higher order and systems of first order equations: $\S 3.3$ Arbitrary constants and initial conditions