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

To determine the correct classification of the given differential equation, let's analyze it step by step.

The given equation is:

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

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, $x$).
   • Here, the highest derivative present is $\frac{dy}{dx}$, which is the first derivative of $y$ with respect to $x$.
   • Therefore, this is a first-order differential equation.

2. Check if the differential equation is separable:

   • A differential equation is separable if it can be written in the form $g(y) dy = h(x) dx$.
   • To check if the given equation is separable, we need to see if we can rearrange it to separate the variables $x$ and $y$.
   • Rearranging the given equation: $\sin(xy) \left(\frac{dy}{dx}\right)^3 = -2x$ $\left(\frac{dy}{dx}\right)^3 = \frac{-2x}{\sin(xy)}$
   • It is not straightforward to separate the variables $x$ and $y$ in this form, so it is not a separable differential equation.

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

   • A third-order differential equation would involve the third derivative of $y$ with respect to $x$, denoted as $\frac{d^3y}{dx^3}$.
   • The given equation only involves the first derivative $\frac{dy}{dx}$, so it is not a third-order differential equation.

4. Determine if it is a second-order differential equation:

   • A second-order differential equation would involve the second derivative of $y$ with respect to $x$, denoted as $\frac{d^2y}{dx^2}$.
   • The given equation does not involve the second derivative, so it is not a second-order differential equation.

5. Check if it falls under "None of the others":

   • Since we have determined that the equation is a first-order differential equation and it does not fit the other categories (separable, second-order, third-order), it does not fall under "None of the others".

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

a. First order differential equation