In calculus, differential equations is one of the major concepts where we learn to equate a derivative. An equation that has one or more terms and derivatives of one variable with respect to another is called differential equation. It is generally represented as:
dy/dx = f(x)
Where x is an independent variable and y is a dependent variable.
Degree and Order
The degree of a differential equation represents the number of times a variable is differentiated. Degree represents the highest order of a given equation. Suppose if y is differentiated once, then dy/dx = y’ have degree equal to 1.
Y’’ + 2y’→ Degree = 2
Y’’’ + 2y’’ + 3y’→ Degree = 3
And so on.
First Order & Second Order
An example of a differential equation is dy/dx = 4x or y’ = 4x. As we can see here, the derivative of variable y has one degree, that means it is differentiated only once. Therefore, such differential equations are called first order differential equations.
Similarly, if the dependent variable is differentiated two times, then the degree of the derivative will be 2. Such an equation with a derivative of 2 degrees is called a second order differential equation.
(d2y)/dy2 = 3x or y’’ = 3x
Types of Differential Equations
There are different types of differential equations we learn in Mathematics. They are:
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations
- Exact Differential Equation
Ordinary & Partial
An ordinary differential equation (ODE) has one function and its derivative. It is the most basic type that has one independent variable and one or more derivatives. Whereas, in contrast to ODE, a partial differential equation (PDE) has unknown multi-variables and their partial derivatives.
Linear & Non-linear
Linear differential equations represent the equation is linear in unknown function and its derivatives, whereas non-linear differential equation is non-linear. So, basically, an equation in the form of dy/dx+Py=Q is a linear equation in first order.
Homogeneous & Non-homogeneous
A homogeneous differential equation is in the form of f(x,y)dy = g(x,y)dx where the degree of f(x,y)dy and g(x,y)dx is the same.
For example, let a function be, F(x,y)=2x−8y
Now if we replace x and y with ux and uy, respectively, where u ≠ 0, then;
F(ux,uy) = 2(ux)−8(uy) = u(2x−8y) = uF(x,y)
A non-homogeneous linear differential equation does not obey the above condition and is represented in the form of:
y”+p(t)y’+q(t)y = f(t), where f(t) is a non-zero function.
Exact Differential Equations
An exact differential equation has the function of exactly two variables with their continuous partial derivatives. Suppose P (x,y) dx + Q (x,y) dy=0 is an exact differential equation such that:
ux(x, y) = p(x, y) and uy(x, y) = Q(x, y).
Therefore, the general equation becomes u(x,y) = C, where C is a constant.
Applications of Differential Equations
The differential equation is used in various fields such as medicine, engineering, architecture, construction, navigation, statistics, etc. Some of the applications are given below:
- To helps to describe exponential growth and decay
- Used in medical field to model the spread of any disease in human body
- To help in business analysis and optimal investment strategies
- To represent movement of electricity