Determining if Systems are Linear

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1. Jan 16, 2017

squelch

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

For each of the following, determine if the system is linear. If not, clearly state why not.
(a) $y''(t)+15y'(t)+sin(y(t)))=u(t)$
(b) $y''(t)-y'(t)+3y(t)=u'(t)+u(t)$
(c) $y'(t)=u(t)$ and $z'(t)=u(t)-z(t)-y(t)$

2. Relevant equations

None

3. The attempt at a solution

Intuitively, I believe that the $sin(y(t))$ term makes (a) nonlinear, while all of the others can be expressed as linear differential equations, but I wasn't sure. I was hoping for a bit of a sanity check on this. Does the introduction of a trigonmetric function like sin() make these nonlinear all of the time, or is it just that the function is nested, e.g. as $sin(y(t))$?

2. Jan 16, 2017

Staff: Mentor

The first equation (a), is nonlinear. A linear differential equation consists of a linear combination of the dependent variable (y(t) in this case) and its derivatives. By "linear combination" I mean a sum of constant multiples of the the dependent variable and its derivatives. Having the sin(y(t)) term makes this equation nonlinear.

3. Jan 16, 2017

Ray Vickson

You are correct: a linear DE is one in which $y(t)$ and all its time-derivatives appear linearly.

That means that we can have coefficients that are functions of $t$ (linear or nonlinear) and still have a linear DE. So, for example, the equation $t^2 y''(t) - 2 t y'(t) + t \sin(t) y(t) = 0$ is still regarded as a linear differential equation.