# Linearity in Differential Equations

1. Mar 26, 2012

### TranscendArcu

1. The problem statement, all variables and given/known data

Is the following differential equation linear:

$yy' + 2 = 0$

3. The attempt at a solution

I have the definition of linear as being $a_0 (t) y^{(n)} + a_1(t) y^{n-1} + a_2 (t) y^{n-2} ... = 0$. Now, presumably y is a function of t. Thus, I could define $y = a_0 (t)$ and let n=1. Thus I would satisfy my definition of linearity in differential equations. Thus, the differential equation is linear.

Is it not so?

2. Mar 26, 2012

### HallsofIvy

No, $a_0(t)$ has to be a known function of t, not y or anything involving y. This equation is non-linear because you have a product of things "involving" the dependent variable y.