Find f. f double prime (theta) = sin theta + cos theta, f(0) = 3, f prime (0) = 4.

Question:

Find {eq}f {/eq}.

{eq}{f}''(\theta) = \sin \theta + \cos \theta, \; f(0) = 3, \; {f}' (0) = 4 {/eq}

Solving Nonhomogeneous Equations Using Undetermined Coefficients:

If we have an initial value problem in the form

{eq}f(D)y = R(x) {/eq}

where {eq}f(D) {/eq} is a differential polynomial, we can find a particular solution {eq}y_p {/eq} by solving the auxiliary equation {eq}g(m) = 0 {/eq}, where g is a polynomial such that

{eq}g(D)R = 0 {/eq}

Our general solution will then be in the form

{eq}y = y_c + y_p {/eq}

where {eq}y_c {/eq} is obtained by solving the auxiliary equation

{eq}f(m) = 0. {/eq}

Answer and Explanation:

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For the initial value problem

{eq}{f}''(\theta) = \sin \theta + \cos \theta, \; f(0) = 3, \; {f}' (0) = 4 {/eq},

the auxiliary equation to the...

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Undetermined Coefficients: Method & Examples
Undetermined Coefficients: Method & Examples

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Chapter 10 / Lesson 15
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The method of undetermined coefficients is used to solve a class of nonhomogeneous second order differential equations. This method makes use of the characteristic equation of the corresponding homogeneous differential equation.


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