anonymous 5 years ago What makes a linear differential equation linear?

1. anonymous

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)

2. anonymous

thank you