Definition of a linear differential equation

Niles
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Homework Statement


Hi all. Is the following definition of a partial linear differential equation correct?


A partial differential equation is called linear if no products of the function and/or its derivatives occur, and if the unknown function and its partial derivatives occur only to the first degree.


I am a little worried about the "and/or" regarding products in the DE. Is it correct that the following term is NOT allowed in a linear differential equation?:

<br /> \frac{\partial u}{\partial x}\frac{\partial u}{\partial t}<br />

- and can you confirm the validity of the "and/or" in my definition? Thanks in advance.

sincerely,
Niles.
 
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Can anybody confirm this?

And also: If I have a function f(x,y)*u, then is the equation still homogeneous and linear? As an example, take f(x,y)=cos(x):
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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