# What makes an ODE linear?

• Aug 27th 2013, 05:53 PM
euphony
What makes an ODE linear?
I'm having trouble understanding this for some reason...am I right about these?

xy' - 4y = 0 is linear.

y' = y - x^3 is linear.

x' + 2x - e^(-x) = 0 is non-linear.
• Aug 27th 2013, 06:29 PM
topsquark
Re: What makes an ODE linear?
Quote:

Originally Posted by euphony
I'm having trouble understanding this for some reason...am I right about these?

xy' - 4y = 0 is linear.

y' = y - x^3 is linear.

x' + 2x - e^(-x) = 0 is non-linear.

An ODE is linear if it is of the form \$\displaystyle a(x)y^{(n)} + b(x)y^{(n - 1)} + ~...~ + c(x)y = F(x)\$. Where \$\displaystyle y^{(n)}\$ is the nth derivative of y(x) and all the a(x), b(x), ...., c(x), F(x) are all functions of x only.

-Dan
• Sep 4th 2013, 06:17 AM
HallsofIvy
Re: What makes an ODE linear?
Note that in the first two you are thinking of "x" as the independent variable- y is a function of x. But in the last, x is the dependent variable- x is a function of some unnamed variable.