Determining if Systems are Linear

[squelch]

Homework Statement

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)##

Homework Equations

None

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))##?

 

[Mark44]

Homework Statement

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)##

Homework Equations

None

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))##?

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.

 

[Ray Vickson]

Homework Statement

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)##

Homework Equations

None

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))##?

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.