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

.

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