# What is the General Solution of the Differential Equation #y''-6y'+10y = 0#?

##### 2 Answers

We have;

# y''-6y'+10y = 0 #

This is a Second Order Homogeneous Differential Equation which we solve as follows:

We look at the Auxiliary Equation, which is the quadratic equation with the coefficients of the derivatives, i.e.

# m^2-6m+10 = 0#

This quadratic does not factorise to I will solve by completing the square (you could equally use the quadratic formula)

# (m-3)^2-3^2+10 = 0#

# :. (m-3)^2 = -1#

# :. m-3 = +-i#

# :. m-3 = 3+-i#

Because this has two distinct complex solutions

# y = e^(pt)(Acosqt+Bsinqt)#

Where

# y = e^(3t)(Acost+Bsint)#

#### Explanation:

The general solution for this kind of differential equation (homogeneous linear with constant coefficients) is

substituting into the differential equation we have

but

with

Using de Moivre's identity

Here