First Order Differential Equation: Overview, Questions, Preparation

In applications, functions typically reflect physically massive numbers, derivatives represent their rates of change, and the difference equation describes a relationship between both. A differential equation is a relation that refers to one or more functions and their derivatives. Thus, differential equations have a leading role in many fields, such as chemistry, physics, economics, and biology.

An equation with only first derivatives is a first-order differential equation, and the one containing the second derivative is a second-order differential equation. An equation dy/dx =f(x,y) describes a first-order differential equation where two variables are x and y. The equation shapes so that there are only a first-order equation and no second-order derivatives.

First Order Differential Equation

When the first-order differential equation y’ = f(x,y) is a linear equation where f is a linear expression in y, the resultant f takes the form of 

f(x,y) = c(x)y + q(x)

Now, consider an equation y’ = f (x,y), which is linear first-order differential equation and can be expressed as: 

y’ + g(x)y = f(x), where f(x) and g(x) are continuous functions of x.

Also, another deduced method to express first-order linear equation is

(dy/dx) + C(x)y = D(x), where C(x) and D(x) are the continuous functions of x.

N.B - When differential equations are in variable separable form, it is easy to solve them and get a viable result.

Methods of solving a first-order differential equation

Following are the two methods of solving a first-order differential equation:
1. Using integrating factor - We know that the general form of a linear first-order differential equation is expressed as:

y’ + g(x)y = f(x)

Now, the integrating factor is expressed as v(x) = exp [∫g(x)dx]
Therefore, solving the first-order differential equation using an integrating factor, which multiplies v(x) to the left side of the equation.

y = ∫v(x).f(x)dx + C/ v(x)

2. Method of constant variation

The first step in this method is to estimate the general equation, which is y’ + g(x)y = f(x).
We must substitute such an unidentified function C for the constant C(x). If this solution is replaced by the non-homogeneous differential equation, the function C(x) can be calculated. The approach to this process is known as the method of variation of the constant.

The topic of the first-order differential equation is from chapter 9 differential equations. Apart from this topic, the chapter also covers order and degree and solutions of homogeneous differential equations of the first order. The chapter is from the unit of calculus, which covers a humongous 35 marks in the 12^th standard final examinations.

1. Determine the order of the given differential equation 2x² d²y/dx - 3 dy/dx + y = 0
Solution
The order of the highest order derivative d²y/dx is 2. Therefore, the given equation is a second-order differential equation.

2. Determine the order of the given differential equation y² + 2y + sin y = 0
Solution
The order of the highest order derivative y² is 2. Therefore, the given equation is a second-order differential equation

Q: Who invented calculus?

Q: Where did Newton first mention about three kinds of differential equations?

Q: What is the primary use of differential equations?

Q: What are the types of differential equations?

Q: How is the order of a differential equation determined?