What is differential equation with examples ?

What is differential equation with examples ?


Differential equation and it's types

Topics covered


1 - what are differential equations ?
2 - what is ordinary differential equation with examples ?
3 - what is partial differential equation with examples ?


Differential equation


An equestion that is consist of derivative of one or more dependent or unknown variables wrt one or more independent or known valuables is called differential equations.

Examples;

1 - dy/dX + 10y = e²
2- d²y/dx² + dy/dX + 8y =0
3 - d⁴y/dx⁴ - d³y/dx³ + d²y/dx² = 0

Types of differential equation


There two basics types of differential equation;

ODE

Ordinary differential equation
PDE

Partial differential equation

Ordinary differential equation


If a differential equation consist of derivative ( ordinary derivative ) of one or more unknown (dependent variables) with respect to a single known (independent variable) then it is called ordinary differential equation.

Examples;

1 – dy/dx – Y = X + 7Y
2 - d⁵y/dx⁵ + X = Y – 4X
3 - d²y/dx² + dy/dx -1 = 0

Partial differential equation


If there is a differential equation that is consist of derivative of one or more dependent variables (unknown) wrt two or more independent variables (known) then it is known as partial differential equation.

Examples;

1 - &²u/&x² + &²u/&y² = 0
2 - &²u/&x² = &²u/&t²

As it clear from name partial differential equation that it is consist of partial derivatives.
Here the sign & represent partial sign.

Classification of differential equation by order


Order of differential equation


The order of highest derivative in ordinary differential equation or partial differential equation is called order of differential equation.(ordinary or partial)

Examples;

1 - d²y/dx² - (dy/dx)⁴ + y = 2x

Here order of ODE is two because it consist of second order dy²/dx² but not (dy/DX)⁴.

2 - &³t/&x³ + &²t/&y² + (&t/&z)⁴ = 0

Here order of PDE is three because it highest order is 3 but not highest power that is 4.

Notations for differential equations


Differential equation can be written by using Leibniz notation such as dy/dX, d²y/dx², .... Or it can be written by using prime notations such as y’,y”,y”’,... But there is a limit to use prime because we can represent first, second and third order differential equations by prime notations and higher derivative differential equations can be written as y⁴ , y⁵ ,...

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