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All Differential Equations Resources

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Flashcards: Undetermined Coefficients

Find the general solution for $y'' - 3y' - 4y = -25\cos(2t)$ .

$y =\frac{3}{2}\sin(3t) + 2\cos(3t)$

$y = 2\sin(3t) + \frac{3}{2}\cos(3t)$

$y = c_1e^{4t} + c_2e^{-t} + \frac{3}{2}\sin(3t) + 2\cos(3t)$

$y = c_1e^{4t} + c_2e^{-t} + 4\cos(3t)$

$y = c_1e^{4t} + c_2e^{-t} + 2\sin(3t) + \frac{3}{2}\cos(3t)$

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Flashcards: Undetermined Coefficients

Find the general solution for $y'' - 3y' - 4y = -25\cos(2t)$ .

Your answer:

$y =\frac{3}{2}\sin(3t) + 2\cos(3t)$

Your answer:

$y = 2\sin(3t) + \frac{3}{2}\cos(3t)$

Your answer:

$y = c_1e^{4t} + c_2e^{-t} + 4\cos(3t)$

Your answer:

$y = c_1e^{4t} + c_2e^{-t} + 2\sin(3t) + \frac{3}{2}\cos(3t)$

Correct answer:

$y = c_1e^{4t} + c_2e^{-t} + \frac{3}{2}\sin(3t) + 2\cos(3t)$

Explanation:

This is a higher order inhomogeneous linear differential equation. Because the inhomogeneity is a cosine, we will use variation of parameters to solve it.

First, we find the characteristic equation to solve for the homogenous solution. This gives us $\lambda^2 - 3\lambda - 4 = (\lambda -4)(\lambda + 1) = 0$ .

This tells us that the homogeneous solution is $c_1e^{4t} + c_2e^{-t}$ . As neither of these overlap with our inhomogeneity, we are safe to continue without adding a factor of t.

Thus, let us guess that $y_p = a_1\sin(2t) + a_2\cos(2t)$ . Then,

$y_p' = 4a_1\cos(2t) - 4a_2\sin(2t)$ and

$y_p'' = -4a_1\sin(2t) - 4a_2\cos(2t)$

Plugging into the original equation, we have

$\sin(2t)(-4a_1 + 6a_2 -4a_1) + \cos(2t)(-4a_2 - 6a_1 -4a_2) = -25\cos(2t)$

Which implies that -8a_1 + 6a_2 = 0 and -8a_2 - 6a_1 = -25 . Solving via substitution,

$a_1 = \frac{3}{4}a_2$

$-8a_2 - \frac{9}{2}a_2 = -\frac{25}{2}a_2 = -25; a_2 = 2$

$a_1 = \frac{3}{2}$

Thus, the particular solution is $y_p = \frac{3}{2}\sin(2t) + 2\cos(2t)$ and the overall solution is the particular plus the homogeneous.

$y = c_1e^{4t} + c_2e^{-t} + \frac{3}{2}\sin(3t) + 2\cos(3t)$

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Flashcards: Undetermined Coefficients

Flashcards: Undetermined Coefficients

Flashcards: Undetermined Coefficients

Flashcards: Undetermined Coefficients

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All Differential Equations Resources

Flashcards: Undetermined Coefficients

Flashcards: Undetermined Coefficients

Flashcards: Undetermined Coefficients

Flashcards: Undetermined Coefficients

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DMCA Complaint

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Complaint Details

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