Week 9

Tutorial 9: Laplace Transform 2

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TUTORIAL 9 LAPLACE TRANSFORM 2.pdf

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Evaluate the given Laplace transform. i.

ii.

iii.

iv.

vi.

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Solution

Question (i)

Question (ii)

Question (iii)

Question (iv)

Question (v)

Question (vi)

Use the Laplace transform to solve the initial-value problem.

ii. where

iii.

iv.

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Solution

Question (i)

Let

Question (ii)

The Laplace transform of the differential equation is

Question (iii)

The Laplace transform of the differential equation is

Question (iv)

The Laplace transform of the differential equation is

Let

Question (v)

Use the Laplace transform to solve the given system of differential equations.

ii.

iii.

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Solution

Question (i)

From , and subsitute into

By partial fraction,

Therefore,

Question (ii)

Let

By partial fraction,

Therefore,

Question (iii)

Then,

Two masses and are connected to three springs of negligible mass having spring constants and , respectively.

Let and represent displacements of masses and from their equilibrium positions. The motion of the coupled system is represented by the system of second-order differential equations:

Using Laplace transform to solve the system when and .

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Solution

For

Substitute unknowns:

Laplace transform:

For

Substitute unknowns:

Laplace transform:

Hence,

From

Substitute into

Inverse Laplace,

The system of differential equations for the charge on the capacitor and the current in the electrical network shown below is

Find the charge on the capacitor using Laplace transform when .

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Solution

Solving and :