MIT 1803 lecture 13

This lecture covered finding the y_p , a particular solution of the ODE

y'' + Ay' + By = f(x)

Let the solutions be generally complex so that e^{(a+iw)x} = e^{\alpha x} .   Express the ODE as

D^2y + A D y + B y = f(x)

let p(D) be (D^2 + AD + B) or any polynomial operator on D , then

p(D) e^{\alpha x} = p(\alpha) e^{\alpha x}

which we will call the substitution rule.  This yields the exponential input theorem

y_p = \frac{e^{\alpha x}}{p(\alpha) }

when p(\alpha) \neq 0 .   If p(\alpha) = 0 , then the exponential shift rule can be used

p(D) e^{\alpha x} u(x) = e^{\alpha x} p(D + \alpha) u(x)

In proving this rule – he again warns you not to hack your way forward – generating more work than necessary – make sure to use your previous result.

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