Particular Solution of Constant Coefficient Linear nth Order ODE

Theorem

Consider the linear $n$th order ODE with constant coefficients:

$(1): \quad \ds \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$

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

$(2): \quad y = y_0, \dfrac {\d y} {\d x} = y_1, \dotsc, \dfrac {\d^{n - 1} y} {\d x^{n - 1} } = y_{n - 1}$

when $x = x_0$.

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

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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