Question

A first- order and first- degree differential equation in variables $x$ and $y$ is characterized by
A. First order of differentiation of $y$ with respect to $x$ and first degree of $\dfrac{{dy}}{{dx}}$
B. Order of differentiation of $y$ with respect to $x$ and power of $y$
C. Powers of $x$ and $y$
D. None of the above

Answer 1

Hint: A differential equation is a mathematical equation that connects some function to its derivatives. In real-world applications, functions represent physical quantities, whereas derivatives represent the rate of change of the function with respect to its independent variables.

Complete step by step solution: 
As we know that,
The highest order derivative involved in a differential equation is referred to as the differential equation's order. The degree of a differential equation is the exponent or power of the highest order derivative in that differential equation.
A first-order differential equation is an equation in which $\left( {x,y} \right)$ is a function of two variables and it can be defined on a $xy - $plane region. As, it only involves the first derivative $\dfrac{{dy}}{{dx}}$, therefore the equation is of first order.
Also, a first- order and first- degree differential equation can be written as
$f\left( {x,y,\dfrac{{dy}}{{dx}}} \right) = 0$
Hence option (A) is the correct answer i.e., first order of differentiation of $y$ with respect to $x$ and first degree of $\dfrac{{dy}}{{dx}}$.

Note: The term derivative is the root of the first order differential equation. A solid understanding of derivatives can make learning differential equations easier and more digestible. A derivative is a mathematical tool for measuring the rate of change of values in a function at a specific point in the function.