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Prove that a given differential equation has a given particular solution

Let L be defined by

    \[ L(y) = y'' + ay' + by, \]

for a, b \in \mathbb{R} constants.

Assume c \in \mathbb{R} and b \neq \omega^2. Prove that the differential equation

    \[ L(y) = c \sin (\omega x) \]

has a particular solution of the form

    \[ y = A \sin (\omega x + \alpha) \]

and find expressions for A and \alpha in terms of a,b,c, and \omega.


Proof. Incomplete.

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