First Order in Differential Linear Equation

Calculus is a branch of mathematics that uses unknown expressions and derivatives to express rates of change. A popular method to apply calculus is creating a differential linear equation with an unknown expression and its derivative. Analysing and evaluating such equations usually yields data on how various quantities change and insight into why and how those changes occur.

Differential linear equations are solved using various methods like using a direct solution, graphing, and computer calculations. This article introduces the central concepts of first-order differential linear equations, which will be discussed in detail.

Definition of differential linear equations

A mathematical expression containing an unknown function represented by y=F(x), including one or more of its derivatives, is known as a differential linear equation. It is expressed using linear polynomials consisting of an unknown variable and its derivatives. Typically, a differential linear equation is represented as: dy/dx + Qy = T

where

• y is an unknown variable.
• dy/dx is the derivative of y. It is also represented as y’.
• Q and T are either functions of variable x or constants.

The highest order of any derivative of the unknown function, y = F(x) appearing in the equation, is defined as the order of a linear differential equation. For example, consider the following differential linear equation: 

dy/dx = 5x².

It includes a single derivative where the derivative of y is on the LHS. As a result, we may deduce the following from this equation:

• An unknown function exists in the form y = F(x). The character ‘F’ denotes the expression ‘is a function of x’.
• It is a differential linear equation in y.
• The differential linear equation has a value of 5×2. Here P is zero, and Q is 5×2.
• It is a first-order differential linear equation because the highest derivative is one. It is represented as y’ or dy/dx.

Solving differential linear equations

In calculus, an integrating factor is used to generally solve a differential linear equation. When ‘f’ and its derivatives are substituted into the equation, the solution to the equation is a functional relationship of y=f(x) that solves the differential equation. 

Let’s understand a standard solution of a linear differential equation using integrating factors.

For a differential equation dy/dx + Qy = T, the solution is of the form 

y * (I.F) = (T * (I.F) * dx) + k

Here I.F. stands for integrating factor and takes the value eQ.dx.

An integrating factor (I.F.) is an expression f(x) that is multiplied with the differential equation on both sides. It is used to produce an integral expression for quick solving of differential equations.

The standard solutions of a differential equation can be written using the three easy procedures below.

1. Simplify the differential linear equation by writing it as dy/dx + Qy = T, where T and Q are constant values or functions of x.
2. Determine the I.F (integrating factor) in differential equation using eP.dx.
3. The solution can be written now using the formula y * (I.F) = (T * (I.F) * dx) + k.

Let us briefly understand the above solution with the first-order differential equation below.

We have the equation as dy/dx – y/x = 4x. 

Substituting in general form dy/dx +Qy = T we have values of T = 4x and Q = – (1/x).

This gives us the I.F = e-(1/x).dx. The I.F can be rearranged as e-logx, which is equivalent to 1/x.

Hence the solution now becomes:

y * (1/x) = 4x.(1/x).dx + k

y/x = 4dx + k

y/x = 4x + k

y = 4×2 + xk, which is the final solution for the problem. Here, ‘k’ is the unknown constant.

Conclusion

Since Newton’s time, differential equations have served a significant role in the evolution of mathematics. They’ve also played an essential part in numerous mathematical applications in the physical sciences, engineering and technology. A common type of differential equation is the linear differential equation. The differential linear equation calculates current in an electrical circuit and determines the velocity of a rising or falling item with air resistance. The above article discusses the first order of linear differential equations and how to solve them.