# How do you find all solutions of the differential equation #(d^2y)/(dx^2)=-4y#?

##### 1 Answer

Jan 10, 2017

# y=Acos(2x)+Bsin(2x) #

#### Explanation:

This is a Second Order homogeneous differential Equation with constant coefficients. We can easily find the general equation (GS) of:

# ay''+by'+c=0 #

By looking at the associated Axillary Equation and its roots

# am^2+bm+c=0 # , then:

We have:

# (d^2y)/dx^2 = -4y => (d^2y)/dx^2 + 4y = 0#

So the Axillary equation is:

# m^2+4=0 => m=+-2i #

As we have pure imaginary roots the GS is of the form:

# y=Acos(2x)+Bsin(2x) #