Is This ODE Linear or Non-Linear?

[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.

Thanks in advance for your help
You guys are great.

 

[pbandjay]

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

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) + ... a_n-1(t)y' + a_n(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.

 

[matematikawan]

I believe it is non-linear due to the e^y.