Linea/Non-Linear ODE typing

1. Jan 21, 2010

rpardo

Hey guys,

I have a pretty simple question but I can't seem to find the answer anywhere.
is

y''+xy'-5=(x^2)(e^y)

linear or non-linear

I believe it is non-linear due to the e^y. But i am not too sure.

You guys are great.

2. Jan 21, 2010

pbandjay

Maybe an easy way to think about it is to consider the concept of a linear combination. If f1, f2, ..., fn are given functions, then a linear combination is another function of the form c1f1 + c2f2 + ... + cnfn.

You have y'', y' and y, which are not written as a linear combination in your ODE, therefore it is not linear. Note that in a linear ODE, the coefficients do not have to be constant. Something like y(n) + a1(t)y(n-1) + ... an-1(t)y' + an(t)y = f(t) is also linear.

A linear ODE can also be written in an operator form, for example y'' + 5y' + 3y = 0 <==> (D2 + 5D + 3)y = 0, where D is the derivative operator.

3. Jan 22, 2010