First-Order Differential Equation Formulas
[Tex]\begin{array}{|c|c|c|} \hline \textbf{Type of Equation} & \textbf{General Form} & \textbf{Solution Method} \\ \hline \text{Linear Differential Equations} & \frac{dy}{dx} + p(x)y = q(x) & \text{Use an integrating factor, } \mu(x) = e^{\int p(x) \, dx}, \text{ then solve } \frac{d}{dx}(\mu(x)y) = \mu(x)q(x). \\ \hline \text{Homogeneous Equations} & \frac{dy}{dx} = f\left(\frac{y}{x}\right) & \text{Make the substitution } v = \frac{y}{x}, \text{ then solve the resulting separable differential equation for } v. \\ \hline \text{Exact Equations} & M(x,y) + N(x,y)\frac{dy}{dx} = 0 & \text{Find a potential function } \Psi \text{ such that } d\Psi = Mdx + Ndy, \text{ where } \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}. \\ \hline \text{Separable Equations} & \frac{dy}{dx} = g(x)h(y) & \text{Separate variables and integrate: } \int \frac{1}{h(y)} dy = \int g(x) dx + C. \\ \hline \text{Integrating Factor} & \text{Often used for non-exact linear equations} & \text{Find an integrating factor } \mu(x) \text{ that makes the equation exact, then proceed as with exact equations.} \\ \hline \end{array} [/Tex]
First Order Differential Equation
A first-order differential equation is a type of differential equation that involves derivatives of the first degree (first derivatives) of a function. It does not involve higher derivatives. It can generally be expressed in the form: dy/dx = f(x, y). Here, y is a function of x, and f(x, y) is a function that involves x and y.
It defined by an equation dy/dx = f (x, y) where x and y are two variables and f(x, y) are two functions. It is defined as a region in the xy plane. These types of equations have only the first derivative dy/dx so that the equation is of the first order and no higher-order derivatives exist.
Differential equations of first order is written as;
y’ = f (x, y)
(d/dx)y = f (x, y)
Let’s learn more about First-order Differential Equations, types, and examples of First-order Differential equations in detail below.