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?
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A simple example where the coefficient of the highest order term (y') : is y
Because the co-efficient is not a constant, this equation is non-linear.
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.