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 = Aem1x + Bem2x), 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.