Particular Solution of Constant Coefficient Linear nth Order ODE

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


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

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 $(2)$.