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Flashcards: Higher-Order Differential Equations

Solve the initial value problem y'' - 4y' + 8y = 0 for y(0) = 1 and y'(0) = 2

$y = 2e^{2t}\cos(2t) - 2e^{2t}\sin(2t)$

$y = e^{2t}\cos(2t)$

$y = e^{2t}\sin(2t)$

$y = 4e^{2t} - 2e^{-2t}$

$y = e^{2t}\cos(2t) - e^{2t}\sin(2t)$

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Flashcards: Higher-Order Differential Equations

Solve the initial value problem y'' - 4y' + 8y = 0 for y(0) = 1 and y'(0) = 2

Your answer:

$y = 2e^{2t}\cos(2t) - 2e^{2t}\sin(2t)$

Your answer:

$y = e^{2t}\sin(2t)$

Your answer:

$y = 4e^{2t} - 2e^{-2t}$

Your answer:

$y = e^{2t}\cos(2t) - e^{2t}\sin(2t)$

Correct answer:

$y = e^{2t}\cos(2t)$

Explanation:

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

$\lambda^2 - 4\lambda + 8 = 0$ We then solve the characteristic equation and find that $(\lambda - 2)^2 + 4 = 0; \lambda = 2 \pm 2i$ (Use the quadratic formula if you'd like) This lets us know that the basis for the fundamental set of solutions to this problem (solutions to the homogeneous problem) contains $e^{2t}\sin(2t), e^{2t}\cos(2t)$ .

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, $y = c_1e^{2t}\sin(2t) + c_2e^{2t}\cos(2t)$ . Plugging in our initial condition, we find that 1 = c_2 . To plug in the second initial condition, we take the derivative and find that $y' = 2c_1e^{2t}(\sin(2t) + \cos(2t)) + 2e^{2t}(\cos(2t) - sin(2t))$ . Plugging in the second initial condition yields 2 = 2c_1 + 2 . Solving this simple system of linear equations shows us that

c_1 = 0; c_2 = 1

Leaving us with a final answer of $y = e^{2t}\cos(2t)$

(Note, it would have been very simple to find the right answer just by taking derivatives and plugging in, but this is not overly helpful for non-multiple choice questions)

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All Differential Equations Resources

Flashcards: Higher-Order Differential Equations

Flashcards: Higher-Order Differential Equations

Flashcards: Higher-Order Differential Equations

Flashcards: Higher-Order Differential Equations

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All Differential Equations Resources

Flashcards: Higher-Order Differential Equations

Flashcards: Higher-Order Differential Equations

Flashcards: Higher-Order Differential Equations

Flashcards: Higher-Order Differential Equations

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