The auxiliary equation arising from the given differential equations is:

A.E.: `m^2+60m+500` `=(m+10)(m+50)` `=0`

So `m_1=-10` and `m_2=-50`.

We have 2 distinct real roots, so we need to use the first solution from the table above (*y = Ae*^{m1x} + *Be*^{m2x}), but we use *i* instead of *y*, and *t* instead of *x*.

So `i=Ae^(-10t)+Be^(-50t)`

We could have written this as:

`i=C_1e^(-10t)+C_2e^(-50t)`

or

`i=K_1e^(-10t)+K_2e^(-50t)`

since we have 2 constants of integration. We would be able to find these constants if we were given some **initial conditions**.