Take the second-order differential equation as an example (higher order, etc.): after simplification, it can be transformed into this form called linear differential equation: P(x)y"+Q(x)y'+R(x)y=S(x) (where P(x), Q(x), R(x), s.

No matter how simplified, a nonlinear differential equation containing y or its derivative is a nonlinear differential equation.

Like y'y=y? Although y is not a power, I can change it into y(y'-y)=0 through equivalent transformation, that is, y=0 or y'-y=0, because y and y' are both powers, so it is a linear differential equation. And their coefficients are constant, so they can be called constant coefficient differential equations.

Another example is (sinx)y'-y=0, because the degrees of y' and y are both 1 (excluding the function term containing x), so it is a linear differential equation. The coefficient of y' is sinx, so it is a linear ordinary differential equation with variable coefficients.

Another example is y'y= 1. No matter how simplified it is (for example, dividing y by the past), it can't be in the form that both y' and y degree are 1, so the equation is a nonlinear differential equation.

One more sentence: linear differential equations have analytical solutions, which can be written in the form of resolution function y=f(x). But nonlinear differential equations are hard to say. Generally speaking, some first-order nonlinear differential equations have analytical solutions. However, it is difficult for nonlinear differential equations of second order or above to have analytical solutions.