# Is $\frac{d\left(\tfrac{1}{2}x^2y^2\right)}{dx} = -xy^2$ considered to be a differential equation, and if so does the type have a name?

I have a constraint in the form of a differential equation:

$$\frac{d\left(\tfrac{1}{2}x^2y^2\right)}{dx} = -xy^2 \qquad (1)$$

To avoid possible misunderstanding: the differentiation is specificallly not partial differentiation.

The equation is satisfied if there is the following relation between x and y:
$$y^2=\tfrac{C}{x^4}$$
(Where C is a constant that needs to be determined)

Substituting that relation into the differential equation:

$$\frac{d\left(\tfrac{1}{2} \tfrac{C}{x^2}\right)}{dx} = -\frac{C}{x^3} \qquad (2)$$

So the differential equation (1) implies this relation between x and y:

$$y^2 x^4 = C \qquad (3)$$

(And I can examine to see if it is safe to simplify that to $$y x^2 = C$$)

To provide some context: it's a case where it is known beforehand that a relation between x and y exists, and the objective is to provide a derivation of it.

The differential equation is used to express a constraint. Does this way of using a differential equation with two variables have a name?

(If this usage of differential equations does have a name I can look up information about it.)

• After a substitution of $z = \frac{1}{2}x^2 y^2$, I'd call the equation "separable". Jul 4, 2021 at 20:24
• If you don't like the notation, replace every occurrence of $y$ with $f(x)$. Jul 4, 2021 at 20:29
• After expanding the derivative on the left-hand side, you should get to a first-order separable linear equation which you can solve through a couple standard methods Jul 4, 2021 at 21:35