9. Solving Integro-Differential Equations

An "integro-differential equation" is an equation that involves both integrals and derivatives of an unknown function.

Using the Laplace transform of integrals and derivatives, an integro-differential equation can be solved.

Similarly, it is easier with the Laplace transform method to solve simultaneous differential equations by transforming both equations and then solve the two equations in the s-domain and finally obtain the inverse to get the solution in the t-domain.

Example 1 (Integro-Differential Equation)

Solve the equation for the response i(t), given that

(di)/(dt)+2i+5int_0^ti\ dt=u(t)

and i(0) = 0.

Solution Graph

This is the graph of the solution we just found:

Example 2 (Simultaneous DEs)

Solve for x(t) and y(t), given that x(0) = 4, y(0) = 3, and:

(dx)/(dt)+x+4y=10

x-(dy)/(dt)-y=0

The rectangular plot of the solution is an interesting curve:

The graph of (x(t), y(t)), for -3 < t < 3, showing the point (3, 4) at t = 0.

The curve start at the top (at t = -3) and loops anticlockwise until t=0 (at the point (3, 4), and then the loop gets smaller and smaller, approaching (2,2) as t->oo.