Laplace Transform and solutions for Differential equations.

Cuthbert Nyack

Laplace Transform can be used for solving differential equations by converting the differential equation to an algebraic equation and is particularly suited for differential equations with initial conditions.

The solution requires the use of the Laplace of the derivative:-

Consider the first order differential equation for y(t) below:-

Taking the Laplace Transform (Y(s) is the transform of y(t) and V(s) is the transform of V(t)) and assuming no initial condition gives:-

Solving for Y(s) the Laplace Transform of y(t) gives:-

With a unit impulse input V(s) is given below

and the corresponding Y(s) and y(t) is shown below:-

For a unit step V(S) is shown below

and the corresponding Y(s) is:-

with inverse y(t) given by:-

If y(t) has an initial value of y(0) then the solution contains an additional term y(0)e^-at.

For a unit ramp, the Laplace transform V(s) is

Y(s) is now given by:-

and the corresponding inverse y(t) is:-

If y(t) has an initial value of y(0) then the solution contains an additional term y(0)e^-at.

For second order equations:-
y''(t) + by'(t) + cy(t) = V(t)
Laplace transform gives s²Y(s) + bsY(s) + cY(s) = V(s) + y(0)(s + b) + y'(0)
and Y(s) = V(s)/(s² + bs + c) + y(0)(s + b)/(s² + bs + c) + y'(0)/(s² + bs + c). For any given V(s) the solution can be found but is lengthy because different expressions must be found for the underdamped, critically damped and underdamped cases.
Instead of writing out the expressions, the result is summarised in the following applet which shows solutions of first and second order equations with step, ramp and sine inputs. It also shows the solution of a third order equation with step and sine inputs.
Scrollbar 0 changes the function of the applet and can be changed from 0 to 7 to show 1st, 2nd and 3rd order solutions.
In each case the contribution from the initial conditions are also shown.

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