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Differential equations of the First order and First degree

A differential equation that contains derivative of a function with an exponent one is called the differential equation of the first order and first degree.

Introduction

In calculus, we see several types of differential equations but the first order and first degree differential equations are frequently appeared. So, let us learn how the differential equations of first order and first degree look like.

Example

$(x^2-yx^2)\dfrac{dy}{dx}$ $+$ $y^2$ $+$ $xy^2$ $\,=\,$ $0$

In this example, $x$ and $y$ are variables but $y$ represents a function in terms of $x$. The function $y$ is differentiated one time and there is no other derivatives in the given differential equation. Hence, it is called the first order differential equation.

$\implies$ $(x^2-yx^2)\Bigg(\dfrac{dy}{dx}\Bigg)^1$ $+$ $y^2$ $+$ $xy^2$ $\,=\,$ $0$

The power of the derivative of the function is one. Hence, it is a first degree differential equation.

Therefore, the given differential equation is called the differential equation of first order and first degree.

Examples

The following are some more examples for the first order and first degree differential equations.

$(1)\,\,\,$ $\sec^2{y}\tan{x}\dfrac{dy}{dx}$ $+$ $\sec^2{x}\tan{y} = 0$

$(2)\,\,\,$ $(x-y)^2\dfrac{dy}{dx} = a^2$

$(3)\,\,\,$ $\dfrac{dy}{dx} = -\dfrac{x\sqrt{1-y^2}}{y\sqrt{1-x^2}}$

$(4)\,\,\,$ $\dfrac{dy}{dx}$ $\,=\,$ $e^{2x-3y}+4x^2e^{-3y}$

$(5)\,\,\,$ $x^4\dfrac{dy}{dx}$ $+$ $x^3y$ $+$ $\log_e{(xy)}$ $\,=\,$ $0$