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

Jan 10, 2017

$y = A \cos \left(2 x\right) + B \sin \left(2 x\right)$

#### Explanation:

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

$a y ' ' + b y ' + c = 0$

By looking at the associated Axillary Equation and its roots

$a {m}^{2} + b m + c = 0$, 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:

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - 4 y \implies \frac{{d}^{2} y}{\mathrm{dx}} ^ 2 + 4 y = 0$

So the Axillary equation is:

${m}^{2} + 4 = 0 \implies m = \pm 2 i$

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

$y = A \cos \left(2 x\right) + B \sin \left(2 x\right)$