Differential equations

 Differential Equations

In Mathematics, a Differential equation is an equation that relates one or more unknown functions and their derivatives.
In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

 The Types of Differential Equations

Differential equations can be divided into several types namely :-

1. Ordinary Differential Equations (ODE)

2. Partial Differential Equations (PDE)

3. Linear Differential Equations (LDE)

4.Nonlinear differential equations (NDE)

5. Homogeneous Differential Equations (HDE)

6. Nonhomogeneous Differential Equations(NHDE)

Now , before deep diving 🤿 into the world  of Diffrential equations, let's get familiar with the following terms:-

1. Order of Differential Equation

The order of the differential equation is the order of the highest order derivative present in the equation. 

Examples :-

A)

dy/dx = 3x + 2  

Sol.   The order of the equation is 1

B)

(d³y/dt³)+y = kt

Sol.    The order is 3

2. Degree of Differential Equation

The degree of the differential equation is the power of the highest order derivative.

Examples :

A) (y”’)3 + 3y” + 6y’ – 12 = 0

Sol.     Degree is 3

B) (d²y/dx²) + cos(dy/dx) = 0; 

Sol.    It is not a polynomial equation in y′ and the degree of such a differential equation can not be defined.

3. Differential Equations Solutions

A function that satisfies the given differential equation is called its solution. The solution that contains as many arbitrary constants as the order of the differential equation is called a general solution. The solution free from arbitrary constants is called a particular solution. There exist two methods to find the solution of the differential equation.

• Separation of variables: In the form of dy/dx = f(y)g(x) where f is the function of y only and g is the function of x only. Taking an initial condition, rewrite this problem as 1/f(y)dy= g(x)dx and then integrate on both sides.

• Integrating factor :Integrating factor is defined as the function which is selected in order to solve the given differential equation. It is most commonly used in ordinary linear differential equations of the first order. When the given differential equation is of the form; dy/dx + P(x) y = Q(x)

4. Linear Equation: A linear equation is an equation in the form,

L(u)=f

where 

L is a linear operator & f being the function 

   

 APPLICATIONS OF DIFFERENTIAL EQUATIONS

We are gonna discuss about the basic introduction of ODE which would be followed by a deep analysis on these  topics in further blogs.