8. Using Inverse Laplace Transforms to Solve Differential 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 = lap{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) MATH $G-g(0)$

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


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

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

laplace{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:

laplace{i''(t)} = s2I − s i(0) − i'(0)


(3) For the n-th derivative,

MATH


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

MATH


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.

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


Answer


Scientific Notebook solution:

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

Answer


(b) Solve MATH, given that

y = 1, $\dfrac{dy}{dt}=0$, when t = 0.


Answer


Scientific Notebook solution:


Answer


(c) MATH, given that

y = -2, $\dfrac{dy}{dt}=-3$, when t = 0.


Answer


Scientific Notebook solution:


Answer



APPLICATION

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

MATH

and i(0) = 0, i'(0) = 0.

Determine the current as a function of t.


Answer


Scientific Notebook solution:


Answer




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