Chapter 1: Q48MP (page 1)

Solve Laplace transforms and the convolution integral or by Green functions.
$y'' + y = t \sin t$

Short Answer

Expert verified

The solution of given equation is $y (t) = \frac{t \sin t}{4} - \frac{t^{2} \cos t}{4}$

Step by step solution

Given Information

The expression is $y'' + y = t \sin t$

Green Function

The impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions is known as a Green's function.

Laplace Transform

Consider the equation.

$y'' + y = t \sin t$

Take Laplace transform of both side of above equation.

$L (y'' + y) = L (t \sin t)$

$p^{2} L (y) + L (y) = \frac{2 p}{{(p^{2} + 1)}^{2}} L (y) = \frac{2 p}{(p^{2} + 1) {(p^{2} + 1)}^{2}} y (t) = L^{- 1} \{\frac{2 P}{{(p^{2} + 1)}^{2}} \frac{1}{(p^{2} + 1)}\} y (t) = \int_{0}^{t} τ \sin τ \sin (t - τ) d τ = \int_{0}^{t} τ \sin τ (\sin t \cos τ - \cos t \sin τ) d τ = \frac{\sin t}{t} {[τ (\frac{- \cos 2 τ}{2}) + \frac{\sin 2 τ}{2}]}_{0}^{t} - \frac{\cos t}{2} {[\frac{τ^{2}}{2}]}_{0}^{t} + \frac{\cos t}{2} {[\frac{\cos 2 τ}{4} + τ \frac{\sin 2 τ}{2}]}_{0}^{t} y (t) = \frac{t \sin t}{4} - \frac{t^{2} \cos t}{4}$

Thus, the solution of given equation is $y (t) = \frac{t \sin t}{4} - \frac{t^{2} \cos t}{4}$ .

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Solve Laplace transforms and the convolution integral or by Green functions.
$y'' + y = t \sin t$

Short Answer

Step by step solution

Given Information

Green Function

Laplace Transform

One App. One Place for Learning.

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Solve Laplace transforms and the convolution integral or by Green functions.y''+y=tsint

Short Answer

Step by step solution

Given Information

Green Function

Laplace Transform

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

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Study anywhere. Anytime. Across all devices.

Solve Laplace transforms and the convolution integral or by Green functions.
$y'' + y = t \sin t$