Re: Linearity question, is the proffesor wrong ?

Hi all, I am reading a book on control theory written
by a proffessor
at
Chalmers university of Technology. His name is Bertil
Thomas.

At page 116 he gives two examples of what he think is
nonlinear
differential equations:
2y' + y=5*sqrt(u) and
12y''+6y=5uy+sin(u)

u is an external signal, not yet defined, and is a
function of t, only.
y is the dependent variable, and is a function of t,
the independent
variable.

It is not written as a system of equations, it is
just according to him
two examples of nonlinear differential equations.

According to what I've learned a D.E. is nonlinear
iff the dependent
variable or its derivatives has power not equal to 1,
y^2, y^0.5,
(dy/dt)^3, etc.

I wrote him an email and said to him that he has
mixed things up,
because of he uses the laplacetransform extensively,
and that only
works with differential equations with constant
coefficients, then he
thinks that if a D.E. has noncnstant coefficients it
is non-linear.

He responded that there are many definitions of
linearity, and that he
uses the definition of superposition of two input
signals.

I asked around, and got the answer that thati s the
only definition of
linearity for D.E.

Is the proffessor correct ?
Why can't we use the laplacetransform on D.E. with
non-constant
coefficients.

Kindest regards,
Lasse

If I were you I would have NOT simply written telling him he is wrong! (Especially, if I were taking the course for credit!) I might have written him asking for clarification. I am no expert in "control theory" but since this is a book on that subject, it might well use the phrase "non-linear problem" to that it does not act "linearly" on the inputs. I might prefer that the book not call them non-linear "equations". Does he actually do that rather than saying non-linear "problems"?
.

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