Question

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

Answer 1

`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:

#{ ("real distinct roots", alpha","beta,=>y,=Ae^(alpha x) + Be^(beta x)), ("complex roots", p+-qi,=>y,=e^(px)(Ae^(qix)+Be^(-qix))), ("",,,=e^(px)(Ccos(qx)+Dsin(qx))), ("pure imaginary roots", +-qi,=>y,=Ccos(qx)+Dsin(qx)), ("equal roots", alpha,=>y,=(A+Bx)e^(-alpha x) ):} #

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)`