# 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:

The graph of *i*(*t*).

### 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`.

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