The equationsin(xy)(dydx)3+2x=0sin⁡(𝑥𝑦)(𝑑𝑦𝑑𝑥)3+2𝑥=0is a Question 38Select one:a.None of the othersb.Separable differential equationc.Third order differential equationd.Second order differential equatione.First order differential equation

To determine the type of differential equation given by the equation:

$\sin(xy) \left(\frac{dy}{dx}\right)^3 + 2x = 0$

we need to analyze its components and structure. Let's break it down step by step:

1. Identify the order of the differential equation:

   • The order of a differential equation is determined by the highest derivative of the dependent variable (in this case, $y$) with respect to the independent variable (in this case, $x$).
   • In the given equation, $\left(\frac{dy}{dx}\right)^3$ is present. The term $\frac{dy}{dx}$ represents the first derivative of $y$ with respect to $x$.
   • Since the highest derivative present is $\frac{dy}{dx}$, which is the first derivative, the equation is a first-order differential equation.

2. Check if the equation is separable:

   • A separable differential equation can be written in the form $g(y) \, dy = f(x) \, dx$.
   • The given equation $\sin(xy) \left(\frac{dy}{dx}\right)^3 + 2x = 0$ does not easily separate into a product of functions of $x$ and $y$ on opposite sides of the equation. Therefore, it is not a separable differential equation.

3. Determine if it is a higher-order differential equation:

   • A second-order differential equation involves the second derivative $\frac{d^2y}{dx^2}$.
   • A third-order differential equation involves the third derivative $\frac{d^3y}{dx^3}$.
   • Since the given equation only involves the first derivative $\frac{dy}{dx}$, it is neither a second-order nor a third-order differential equation.

Based on this analysis, the correct classification of the given differential equation is:

e. First order differential equation