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Chapter Contents

  Laplace Transforms   1a. The Unit Step Function - Definition   Oliver Heaviside   1b. The Unit Step Function - Products   2. Laplace Transform Definition   Table of Laplace Transformations   3. Properties of Laplace Transform   4. Transform of Unit Step Functions   5. Transform of Periodic Functions   6. Transforms of Integrals   7. Inverse of the Laplace Transform   8. Using Inverse Laplace to Solve DEs  9. Integro-Differential Equations and Systems of DEs   10. Applications of Laplace Transform   Laplace Transforms Problem Solver 

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8. Using Inverse Laplace Transforms to SolveDifferential Equations

Laplace Transform of Derivatives

We use the following notation:

Later, on this page...

Subsidiary Equation

Application

(a) If we have the function g(t ), then G(s) = G = {g(t )}. 

(b) g(0) is the value of the function g(t ) at t = 0.

(c) g' (0), g'' (0),... are the values of the derivatives of the

function at t = 0. 

If g(t ) is continuous and g'(0), g''(0),... are finite, then

(1)

(2) {g''(t )} = s2G − s g (0) − g' (0) 

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We saw many of these expressions in the Table of LaplaceTransforms. 

NOTE: If instead of g(t ) we have a function y of x, thenEquation (2) would simply become:

{ y'' ( x)} = s2Y − s y (0) −  y' (0) 

Likewise, if we have an expression for current i and it is a

function of t , then the equation would become:

{i'' (t )} = s2 I − s i(0) − i' (0) 

(3) For the n-th derivative,

NOTE: If we have y and it is a function of t , then thenotation would become:

Subsidiary Equation

The subsidiary equation is the equation in terms

of s, G and the coefficients g' (0),g'' (0),... etc ,obtained bytaking the transforms of all the terms in a linear differentialequation.

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The subsidiary equation is expressed in the form G = G(s). 

EXAMPLES

Write down the subsidiary equations for the followingdifferential equations and hence solve them.

(a) , given that y = 0 when t = 0. Answer 

Taking Laplace transform of both sides gives:

(since y(0) = 0)

Solving for Y and finding the partial fraction decompositiongives:

gives gives

gives gives

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gives gives

So

Finding the inverse Laplace tranform gives us the solutionfor y as a function of t :

Scientific Notebook solution:

This is the way we could go about this problem using ScientificNotebook. You don't need any special software to see it (it is in HTMLform.)

Answer 

(b) Solve , given that

 y = 1, , when t = 0. 

Answer 

Taking Laplace transform of both sides and appying initial

conditions of y(0) = 1 and y' (0) = 0 gives:

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 Solving for Y and completing the square on thedenominator gives:

Now, finding the inverse Laplace Transform gives us thesolution for y as a function of t :

Scientific Notebook solution:

Answer 

Using Scientific Notebook, we can solve it in one step.

We set up a matrix with the differential equation and initialconditions:

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Exact solution is:

Note: In SNB , we can also do "Laplace solution". It gives

the same answer.

The graph of what we have found:

(c) , given that

 y = -2, , when t = 0. 

Answer 

Taking Laplace transform of both sides:

Applying the initial condition and simplifying gives:

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Solving for Y :

For the first term, we use: ,

with a = 1 and n = 2. 

So

For the second term, we express in partial fractions:

Comparing coefficients:

gives A = -2.

gives B = -1.

So

And

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Putting our inverse Laplace transform expressions together,the solution for y is:

Scientific Notebook solution:

Answer 

APPLICATION

The current i(t ) in an electrical circuit is given by the DE

and i(0) = 0, i' (0) = 0. 

Determine the current as a function of t .

Answer 

We need to write the RHS of the DE in terms of unit step

functions.

Now, taking Laplace transform of both sides gives us:

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We need to find Inverse Laplace. First, we concentrate on

the part and ignore the part fornow.

Now

gives 1 = 2 B gives

gives 1 = 4C gives

gives 1 = 3 A + 3 B + C gives

So

So since

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then we have, using the Time-Displacement Theorem (see

the Table of Laplace Transforms):

The graph of i(t ) is as follows:

Scientific Notebook solution:

Answer 

7. Inverse of the Laplace Transform

9. Integro-Differential Equations and Systems of DEs

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