# Homework Help: Differential equation linear input/output

1. Jan 24, 2010

### gomerpyle

1. The problem statement, all variables and given/known data

Given the equation my'' + cy' + ky = ku + cu' explain why the size of the output scales linearly with the size of the input. In your argument, do NOT actually try to find the solution(s) to the equation, rather use the definition of what it means to be a solution to a DE in your explanation.

2. Relevant equations

my''(t) + cy'(t) + ky(t) = ku(t) + cu'(t)

This equation models a physical system where m is a mass, c is a damper, and k is a spring. y is the output and u is the input.

3. The attempt at a solution

I'm probably reading too much into this, given the equation is not very simple but the explanation supposedly should be pretty easy.

I know the solution is made up of the homogeneous and particular parts. Since the particular solution is the response to input, I'm guessing the explanation has something to do with that. The solution y(t) should also satisfy the equality in the equation for all values of t in order to be a solution. Must y(t) = u(t) and thus increase u(t) by a factor of some constant will thus affect the output?

I've done searches on linear input/output systems, but most sources give lengthy mathematical discussions about why they are linear. Could someone steer me in the right direction please?

2. Jan 24, 2010

### Dick

Well, if u(t) and y(t) solve that equation, and C is some constant, then don't C*u(t) and C*y(t) also solve it? This wouldn't be true if the equation were something like y'(t)^2=u(t), right? I think it's as simple as that.