Differential Equation Formula ( Definition, Formula, List & Examples)

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About Differential Equation Formula

A differential equation is an equation that contains the dependent and independent variables, as well as the derivatives of the dependent variable. Ordinary differential equations are sometimes known as differential equations with only one independent variable. The order of a differential equation can be defined as the order of the differential equation's highest derivative.

These equations are utilised in a wide range of fields, including physics, chemistry, biology, geology, and economics. As a result, a thorough understanding of differential equations is critical in all modern scientific investigations. An ordinary differential equation is a differential equation with only one independent variable and derivatives of the dependent variable. Get the list of Maths Formulas.

Some examples based on Differential Equation Formula

For some function g, find another function f such that

$\frac{dy}{dx} = f (x)$

where $y = f (x)$

This is the differential equation. As a result, a differential equation is defined as an equation in which the dependent variable's derivatives or derivatives concerning the independent variable are included.

Some examples of differential equations:

$\frac{dy}{dx} = xsinx$
$\frac{dy}{dx} = ysinx$
$\frac{dy}{dx} = xsiny$
$\frac{dy}{dx} = xsiny + ysinx$

Ordinary Differential Equation

An ordinary differential equation is one in which the dependent variable has derivatives concerning only one independent variable. e.g.,

$\frac{2 d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{3} = 0$ is an ordinary differential equation.

Linear Differential Equations

A differential equation of the form:

$\frac{dy}{d x} + My = N$

The first-order linear differential equation, where M and N are constants or functions of x alone, The following are some instances of first-order linear differential equations:

$\frac{dy}{dx} + y = Sinx$

Steps used to solve first-order linear differential equations are:

Write the equation in the form as :
$\frac{dy}{dx} + My = N$
where M and N are constants or functions of x only.
Find the Integrating Factor (I.F)
^{$I . F . = e^{\int Pdx}$}
Write the solution as:
$y (I . F .) = \int (Q \times I . F .) dx + C$

If the first-order linear differential equation is:

_{$\frac{dx}{dy} + M_{1} x = N_{1}$}

where and N₁ are constants or functions of y only. Then

^{$I . F . = e^{\int M_{1} dy}$}

and the solution to it will be given by:

$x . (I . F .) = \int (N_{1} \times I . F .) dy + C$

Also, $\frac{dy}{dt} + P (t) y = g (t)$ is also a type of differential equation.

P(t) & g(t) are the functions.

$Y (t) = \int μ (t) g (t) d (t) + cμ (t)$

Where $μ (t) = e \int P (t) d (t)$

Differential Equation Formula

About Differential Equation Formula

Some examples based on Differential Equation Formula

Ordinary Differential Equation

Linear Differential Equations

Download the pdf of the Differential Equation Formula

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