Question

What makes a linear differential equation linear?

Answer 1

A differential equation is linear if it can be written in the form,

\[
a_{n}(t) y^{(n)} + a_{n-1}(t) y^{(n-1)}+\cdots+a_{1}(t) y^{\prime}+a_{0}(t) y=g(t)
\]

where \( y^{(n)}\) is the nth derivative of y.

Or, another way to put is it is that the function y and it's derivatives need to be in terms by themselves (i.e. no products/quotients etc of the y or it's derivatives). The function and it's derivatives need to be in the numerator with exponents of 1 and can't be "inside" other functions (i.e. no \( cos(y^{\prime})\), \(e^{y}\), etc)

Answer 2

thank you