A differential equation is given along with the field or problem area in which it arises. Classify it as an ordinary differential equation (ODE) or a partial differential equation (PDE), give the order, and indicate the independent and dependent variables. If the equation is an ordinary differential equation, indicate whether the equation is linear or nonlinear.
[y + (dy/dx)²] = C ,
where C is a constant (brachistochrone problem, calculus of variations).
Classify the given differential equation. Choose the correct answer below:

A. linear ordinary differential equation
B. nonlinear ordinary differential equation
C. partial differential equation

As this equation contains derivatives and not partial derivatives, this equation is an ordinary differential equation and not a partial differential equation.

Here, [tex]\frac{\mathrm{d} y}{\mathrm{d} x}[/tex] denotes that x is an independent variable and y is a dependent variable.

Order of the differential equation is a number of the highest derivative.

Order of the given differential equation is 1.

We can write this ordinary differential equation as follows:

[tex]y+\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^2=C\\\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^2=C-y\\\frac{\mathrm{d} y}{\mathrm{d} x}=\sqrt{C-y}[/tex]

As this equation is of form [tex]\frac{\mathrm{d} y}{\mathrm{d} x}=f(x,y)[/tex], this is a linear differential equation.

So, option A. is correct as this is a linear ordinary differential equation.