Differential equation is an equation that includes the derivatives of a function as well as the function itself. If partial derivatives are concerned, the equation is known as partial differential equation; if only ordinary derivatives are available, the equation is known as ordinary differential equations play a very significant and useful role in applied math, engineering, and physics, etc. Let, us see important types of differential equation and its definition.
Ordinary Differential Equations:
An Ordinary Differential Equation is one of the type of differential equation that depends on only one independent variable.
Here `dy/dx=ky(t)` is an Ordinary Differential Equation since y (the independent variable) based only on t(the independent variable)
Partial Differential Equations
A Partial Differential Equation is another type of differential equation in which the dependent variable based on two or more independent variables.
The Laplace's equation `(del^2f)/(delx^2)+(del^2f)/(dely^2)=0` is a Partial Differential Equation since f based on two independent variables x and y.
Order of a Differential Equation:
The order of a differential is defined as the order of the uppermost derivative entering the equation.
The equation `m(d^2x)/(dt^2)=-kx` is known as a second-order differential equation since it includes second derivatives.
Linear Differential Equation:
A first-order differential equation is linear if it can be defined in the form of `dy/dt+g(t)y=r(t)` here, g(t) and r(t) are arbitrary functions of t.
`dy/dt=t^2y+cos(t)` is called the first-order linear differential equation here` g(t)=t^2` and `r(t)=cos(t)` .
Nonlinear Differential Equation:
It is a differential equation in which the right hand side is not a linear function of the dependent variable.
`(dP)/dt=k(1-P/N)P`
Homogeneous Differential Equation:
A linear first-order differential equation is homogeneous in which right hand side is zero. that is `r(t)=0` .
`dy/dt+g(t)y=0`
`dy/dt+ky(t)=0` , here k is a constant, is homogeneous.
Nonhomogeneous Differential Equation:
A linear first-order differential equation is called non homogeneous in which right-hand side is non-zero that is` r(t)!=0` .
`dy/dt+g(t)y=r(t)`
`dy/dt+2y(t)=sin(2t)` is non homogeneous.
These are the important types of differential eqation.