Second Order Linear Differential Equation

Difference Between Linear and Non-Linear Equations

Last Updated : 21 May, 2024

Difference Between Linear and Non-Linear Equations: While solving mathematical problems, you may have seen types of equations. Few Equations can contain only numbers, others consist of only variables while some consist of both numbers and variables. Linear and nonlinear equations usually consist of numbers as well as variables.

In this article, we will understand the difference between Linear and Non-Linear Equations along with sample problems.

What are Linear and Non-Linear Equations?

Before starting with the difference between linear and non-linear equations, let us first understand the definition of Linear and Non-Linear equations.

Linear Equation

A linear equation is such that it forms a straight line. Linear means something related to a line. All the linear equations are used to construct a line. Linear Equations are conditions of the principal request. These conditions are characterized by lines in the arranged framework. An equation for a straight line is known as a Linear equation. The overall portrayal of the straight-line condition is y=mx+b, where m is the slant of the line and b is the y-catch.

Non-Linear Equation

A non-linear equation is such that it does not form a straight line. It looks like a curve in a graph and has a variable slope value. A non-linear equation is generally given by ax²+by² = c.

where x and y are variables while a,b, and c are constant values.

Difference Between Linear and Non-Linear Equations

The major difference between linear and nonlinear equations is given here for the students to understand it more naturally. The differences are provided in a tabular form with examples:

S No.	Linear Equations	Non-Linear Equations
1.	It forms a straight line or represents the equation for the straight line	It does not form a straight line but forms a curve.
2.	It has only one degree. We can also define it as an equation having a maximum degree of 1.	A nonlinear equation has the degree as 2 or more than 2, but not less than 2.
3.	All these equations form a straight line in the XY plane. These lines can be extended in any direction but in a straight form.	It forms a curve and if we increase the value of the degree, the curvature of the graph increases.
4.	The general representation of a linear equation is; y = mx +c Where x and y are the variables, m is the slope of the line and c is a constant value.	The general representation of nonlinear equations is; ax² + by² = c Where x and y are the variables and a, b and c are the constant values
5.	Linear Equations are much simpler to solve.	Non-linear Equations are tricky in nature.
6.	Linear Equations are time saving.	Non-linear equations are time-consuming.
7.	Examples: 10x = 1 9y + x + 2 = 0 4y = 3x 99x + 12 = 23 y	Examples: x²+y² = 1 x² + 12xy + y² = 0 x²+x+2 = 25

Sample Problems - Difference Between Linear and Non-Linear Equations

Problem 1: Solve the linear equation 3x+18 = 2x + 21.

Solution:

Given, 3x+18 = 2x + 21

⇒ 3x – 2x = 21 – 18

⇒ x = 3

Problem 2: Solve x = 12(x +2)

Solution:

x = 12(x + 2)

x = 12x + 24

Subtract 24 from each side

x – 24 = 12x + 24 – 24

x – 24 = 12x

Simplify

11x = -24

Isolate x, by dividing each side by 11

11x / 11 = -24/11

x = -24/11

Problem 3: Solve the nonlinear equation x+4y = 1 and x = y.

Solution:

Given, x+4y = 1

x = y

By putting the value of x in the first equation we get,

⇒ y + 4y = 1

⇒ 4y = 1

⇒ y = 1/4

∴ x = y = 1/4

Problem 4: Example: Solve the nonlinear equation x+2y = 1 and x = 2

Solution:

Given, x+2y = 1

x = 2

By putting the value of x in the first equation we get,

⇒ 2+ 2y = 1

⇒ 2y = -1

⇒ y = -1/2

∴ y=-1/2

Second Order Linear Differential Equation

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