Linear Ordinary Differential Equation: Definition

[Prof. 27]

Homework Statement

The website says this:
"It is Linear when the variable (and its derivatives) has no exponent or other function put on it.
So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is).
More formally a Linear Differential Equation is in the form:
dy/dx + P(x)y = Q(x)"

My question is whether what makes it a linear differential equation the fact that nothing on the other side of the equals sign from Q(x) has any degree higher than one or whether it is a linear differential equation because the differential doesn't have a degree higher than 1; for example, is this a linear differential equation?
dy/dx + y^3 = Q(x)
What about this one?
dy/dx + P(x)y = Q(x)^2

Homework Equations

The Attempt at a Solution

http://en.wikipedia.org/wiki/Linear_differential_equation
http://www.mathsisfun.com/calculus/differential-equations.html

Note: Sorry on the title. I meant Linear Ordinary Differential Equation. Partial should not be in there.

 

[Mark44]

Homework Statement

The website says this:
"It is Linear when the variable (and its derivatives) has no exponent or other function put on it.
So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is).
More formally a Linear Differential Equation is in the form:
dy/dx + P(x)y = Q(x)"

Maybe a better way to define a linear differential equation is that it will consist only of a linear combination of the dependent variable and its derivatives. In other words, an equation that consists of either a sum of constant multiples of y, y', y'', etc. or a sum where y, y', y'' etc. are multiplied by function of the independent variable.

My question is whether what makes it a linear differential equation the fact that nothing on the other side of the equals sign from Q(x) has any degree higher than one or whether it is a linear differential equation because the differential doesn't have a degree higher than 1; for example, is this a linear differential equation?
dy/dx + y^3 = Q(x)

No, not linear, because of the y³ term.

What about this one?
dy/dx + P(x)y = Q(x)^2

Yes, linear. It doesn't matter that y is multiplied by P(x) or that we have [Q(x)]².

Homework Equations

The Attempt at a Solution

http://en.wikipedia.org/wiki/Linear_differential_equation
http://www.mathsisfun.com/calculus/differential-equations.html

Note: Sorry on the title. I meant Linear Ordinary Differential Equation. Partial should not be in there.

I removed "Partial" from the thread title.

 

[Prof. 27]

Thanks for the help Mark44. I understand now.