# 8. Using Inverse Laplace Transforms to Solve Differential Equations

## Laplace Transform of Derivatives

We use the following notation:

Subsidiary Equation

Application

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

### First Derivative

ℒ{g"'"(t)}= ℒ{(dg)/(dt)} =sG-g(0)

### Second Derivative

ℒ{g"''"(t)}=s^2G-s\ g(0) - g"'"(0)

We saw many of these expressions in the Table of Laplace Transforms.

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

ℒ{y’’(x)} = s^2Y − 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:

ℒ{i’’(t)} = s^2I − s\ i(0) − i\ ’(0)

### For the n-th derivative

ℒ{(d^ng)/(dt^n)}=s^nG-s^(n-1)g(0)-s^(n-2)g ’ (0)-...-g^(n-1)(0)

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

ℒ{(d^ny)/(dt^n)}=s^nY-s^(n-1)y(0)-s^(n-2)y ’ (0)-...-y^(n-1)(0)

## Subsidiary Equation

The subsidiary equation is the equation in terms of s, G and the coefficients g'(0), g’’(0),... etc., obtained by taking the transforms of all the terms in a linear differential equation.

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

## Examples

Write down the subsidiary equations for the following differential equations and hence solve them.

### Example 1

(dy)/(dt)+y=sin\ 3t, given that y = 0 when t = 0.

#### Solution Graph for Example 1

This is the graph of the solution we obtained in the example above.

### Example 2

Solve (d^2y)/(dt^2)+2(dy)/(dt)+5y=0, given that y = 1, and (dy)/(dt)=0, when t = 0.

#### Solution Graph for Example 2

Here is the graph of what we just found:

### Example 3

(d^2y)/(dt^2)-2(dy)/(dt)+y=e^t, given that y = -2, and (dy)/(dt)=-3 when t = 0.

## Application

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

(d^2i)/(dt^2)+2(di)/(dt)=0,if 0 < t < 10

=1,if 10 < t < 20

=0,if t > 20

and i(0) = 0, i’(0) = 0.

Determine the current as a function of t.

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