Question

how to get a order/degree/linearity/unknown function and independent variables of (y')^2-3 y y'+xy=0

Answer 1

its partial ^_^

Answer 2

since u have xy together

Answer 3

\(\bf{Order}\) is given by the highest order derivative. In this case, the highest derivative is the first, so the DE is first order.

\(\bf{Degree}\) is the highest power of the highest order derivative. \(y'\) is squared, so the degree is 2.

\(\bf{Linearity}\) is determined much like the linearity of an algebraic equation.
\[f(x)y'-g(x)y=h(x)\]
is a linear equation (with order and degree 1 - this is a necessary condition for linearity). Compare with the algebraic equation
\[ax+b=c\]
The algebraic equation will be non-linear if it contains a function of \(x\). Examples:
\[ax^2+b=c\\
a+\frac{b}{x}=c\\
a+bx=ce^x\]
Similarly, a differential equation will be non-linear if t contains a function of the derivative or the dependent variable. Examples:
\[yy'=0\\
(y')^2=y\\
y'-\sin y=x\]
Your equation is non-linear.

To determine \(\bf{variable~in/dependence}\): a differential equation can be considered a multivariable function:
\[f(x,y,y',...,y^{(n)})=0\]
Generally, the first variable is the independent variable, the second is the dependent variable, and every variable listed after the dependent is a derivative of the dependent variable.

More info here: http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html