Can Q(x) in a linear differential equation depend on both x and y?

[tascja]

First Order Linear Equation
I have a question about differential equations... The equation for a general linear differential equation that is:

dy/dx + P(x)y = Q(x)

So my question is can you have a Q(x) that has both x and y variables?

For Example:
dy/dx + (1/x)y = (1/x)y^2

 

[snipez90]

Well Q(x) is a function of x, but if you mean ask if the right hand side can have both x and y variables, then yes. The trick is to reduce a nonlinear equation to a first order linear equation. This should help:

http://en.wikipedia.org/wiki/Bernoulli_differential_equation

 

[HallsofIvy]

First Order Linear Equation
I have a question about differential equations... The equation for a general linear differential equation that is:

dy/dx + P(x)y = Q(x)

So my question is can you have a Q(x) that has both x and y variables?

For Example:
dy/dx + (1/x)y = (1/x)y^2

This example is obviously not linear because of the "y²".

If you have dy/dx+ P(x)y= Q(x,y) then there are two possiblities:
a) That the right side is not linear in y so the d.e. is not a linear equation.

b) That the right side is linear. In that case, it is of the form a(x)y+ b(x) and the whole equation can be written dy/dx+ P(x)y= a(x)y+ b(x) or dy/dx+ (P(x)-a(x))y= b(x) which is just the original for again.

So the answer to your question is "no". If your differential equation is linear, then it can be written in that form where Q(x) does NOT depend on y (which was the reason for calling it "Q(x)" to begin with).