Verified answer

See example 4 on pg. 4:

http://www.stewartcalculus.com/data/CALCULUS%20Con...

Good luck.

You'll need to use the quadratic equation to find the complex roots, then afterward use those initial values to find c1 and c2.

You should recognize this as a 2nd order, linear, normal, homogeneous differential equation with constant coefficients.

If not, read about them.

You should know that you need the roots of the characteristic equation.

They are complex, leading to two independent solutions of form e^kt*cos(bt) and same with sine function, where the numbers have to do with the roots in an easy way.

The special conditions lead to the two coefficients in front of the two functions.

You should also know about the webpage listed below. I entered "solve 4y''-24y'+85y=0" and got your answer (not special case stuff)