Question

can anyone explain me the form of first and higher order linear differential equation? I know the form of linear differential equation. e.g.
y''-xy'+2y=0. So what will happen to this equation if the coefficient of y' is y?

Answer 1

A simple example where the coefficient of the highest order term (y') : is y
\[y\cdot y'=a\]

Because the co-efficient is not a constant, this equation is non-linear.

Answer 2

While you are correct with your example being non-linear, in general the non-constancy of coefficients is insufficient to classify an ODE as non-linear.

As the poster noted, their ODE was a linear ODE even though the coefficient on the y' term is a function of x namely f(x)=-x.

A better way of stating it is that as long as the "coefficient" (aka the function) multiplying each term is not a function of the independent variable (y) or its derivatives, then the ODE is linear. Otherwise the ODE is non-linear. The example you provided is precisely such a term that would make an ODE non-linear. Other examples of non-linear terms y^2 or (y')^2.