Use the Laplace transform to solve the following initial value problem: 2y"+4y'+17y=3cos(2t) y(0)=y'(0)=0 a)take Laplace transform of both sides of th

Question

Use the Laplace transform to solve the following initial value problem:

2 y " + 4 y^{'} + 17 y = 3 \cos (2 t)

y (0) = y^{'} (0) = 0

a)take Laplace transform of both sides of the given differntial equation to create corresponding algebraic equation and then solve for

L {y (t)}

b) Express the solution

y (t)

in terms of a convolution integral

Answer 1

Step 1
Given differntial equation is,

2 y " + 4 y^{'} + 17 y = 3 \cos (2 t)

y (0) = y^{'} (0) = 0

L [y "] = s^{2} L {y (t)} - s y (0) - y^{'} (0)

L [y^{'}] = s L {y (t)} - y (0)

L [\cos (a t)] = \frac{s}{s^{2} + a^{2}}

Taking Laplace transform of equation (1),

2 L [y "] + 4 L [y^{'}] + 17 L [y] = 3 L [\cos (2 t)]

s^{2} L {y (t)} - s y (0) - y^{'} (0) + 4 [s L {y (t)} - y (0)] + 17 L {y (t)} = 3 L {\cos (2 t)}

s^{2} L y (t) + 4 s L y (t) + 17 L y (t) = 3 x \frac{s}{s^{2} + 4}

(s^{2} + 4 s + 17) L {y (t)} = \frac{3 s}{s^{2} + 4}

a) ∴ L {y (t)} = \frac{3 s}{s^{2} + 4} \times \frac{1}{s^{2} + 4 s + 17}

b) y (t) = \int_{0}^{t} [3 \cos (3 w)] \times [e^{- 2 t} \cdot \sin \sqrt{13} w] d w

This is required Laplace transform.

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