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# Nonlinear Equation|Definition & Meaning

## Definition

Any **equation** that is **neither** a linear equation or a **polynomial** with **degree** more than or equal to **two** is referred to as a **nonlinear equation.**

The terms **linear** or **non-linear** are very commonly used in **mathematics.** Anything that follows a **straight-line** trend is called **linear,** while any equation, expression, or process that **can’t be modeled** with the help of a **straight-line** equation is called a **non-linear equation.**

The following **figure** shows some examples of some well-known **non-linear equations: **

**Figure 1: Examples of Non-linear Equation**

Here, in the figure above, on the left is a **simple parabola** centered at the origin (vertex at the origin) that is represented by the **equation y = x ^{2}.** The right plot shows a compound

**polynomial**of

**degree two**defined by the

**equation y = x**

^{2}– 4x.## Explanation of Non-linear Equation

A **mathematical equation** is defined as a claim or hypothesis that the values of **two expressions** are **equal** or **related.** Mathematical **equations** comprise **constants** and **variables** as the use may dictate. Values that never change are **constants.** Unknown numeric values that can be changed are represented by **variables** like **y and x**.

Equations are commonly used by mathematicians in **algebraic operations** to discover solutions, produce graphs, and identify patterns. While working to solve **mathematical problems,** we come across various equations.

Some equations just utilize **variables,** while others only use **numbers,** while still others incorporate **both.** These **numbers** and **variables** are commonly used in linear and **nonlinear equations.**

Something linear is **related** to a **line.** A line is built using all of the **linear equations. **If any equation or process, however, can’t be represented by the equation of a **straight line,** then it’s termed a **non-linear equation. **

A non-linear equation may be thought of as a straight line whose **slope** and **intercept** may also be straight lines on their own. In simpler words, a **non-linear equation** changes the **slope** along its **values.** This phenomenon is illustrated in the figure below.

**Figure 2: Change in Slope of a Non-linear Equation**

In the above figure, the **red arrow** roughly depicts the **direction** and **magnitude** of the slope of curve y = x^{2} – 4x at **different points.**

You must be familiar with numerous sorts of **mathematical equations.** Let’s formally go over the **distinctions** between **linear** and **non-linear equations. Definitions** and **examples** can be used to describe the **differences** between them.

**(a)** Instead of a straight line (like the case of linear equations), a **non-linear equation creates** a **curve** whose **slope continuously changes.****(b)** The non-linear equations **represented** in the form of **polynomials** will always have a **degree of two or greater** but never equal to one since only a straight line has a polynomial degree of one.**(c)** As hinted in (a), a non-linear equation assumes the form of a **curve,** and when the **degree** of the polynomial **rises,** the slope of the **graph’s curve** also **rises.****(d)** The equation ax^{2} + by^{2} = c is frequently used to represent and **model non-linear equations,** where a, b, and c are **constants,** and x and y are **variables.**

## Some Common Examples of Non-linear Equations

Similar to linear equations, **non-linear equations** are often used by expert engineers/economists to plan **projects** and make **forecasts.** A few illustrations or example **applications** of **non-linear equations** are described in the paragraphs as follows:

### Non-linear Regression Analysis or Curve Fitting

Data scientists and **machine learning** engineers employ **regression analysis** as a statistical tool to examine the **relationship** or **correlation** between significant **variables of interest.** Problems that are linear or non-linear can both benefit from such **regression analysis.** It might be applied to perform estimation and projection.

An illustration of a nonlinear equation that might be applied in regression analysis is given below:

**Figure 3: Non-Linear Equation in Regression Analysis**

As shown in the **figure,** the **given data points** have been **modeled** by a non-linear equation y = x^{2}, which is simply a **parabola.**

### Non-linear Logarithmic Scales

A **non-linear** equation with a **logarithmic scale** displays **exponential development** on a **graph.** In a number of disciplines, including **actuarial science,** health, archaeology, **mathematics,** forensic science, **finance,** and geology, experts employ **logarithmic scales.**

Such scales are used to display multiplicative factors or perfect changes on a graph **without distorting** the **data**. Examples of logarithmic scales include the **pH scale,** the **Richter scale,** and the **decibel systems.**

A nonlinear equation for a logarithmic scale is usually of the form z = log_{10}(u). The following figure shows the **effect** of **logarithmic scales** on the visualization of **exponential data.**

**Figure 4: Visualization of Exponential Data using Non-Linear Logarithmic Equations**

### Non-linear S-curves

**S-curve graphs** are frequently used in **project management** to define and track the scope and **advancement** of a **project.** They can be helpful in establishing links between certain occurrences and **production pauses** or an increase in product flaws.

S-curves are usually **compound equations** that contain the addition of one or more linear, non-linear, or both types of **equations.** The following straightforward non-linear S-curve equation can be used as a perfect example of such curves:

**y = x ^{2} – 4x**

As you may notice, it is the **same curve** we drew in **figure** **1**.

## Methods for Solving Non-linear Equations

**Polynomials** with more than one degree can be coupled to **form nonlinear algebraic equations,** commonly known as **polynomial equations.** Root-finding algorithms, **for example,** can be used to find solutions to a single **polynomial problem** (i.e., sets of values for the variables that satisfy the equation).

The study of **harder systems** of algebraic equations is one of the goals of **algebraic geometry,** a challenging subject of **modern mathematics.** Even determining if an algebraic system includes **complex solutions** can be challenging.

However, there are many **efficient techniques** for **solving polynomial equation systems** with a finite number of **complex solutions.**

Some commonly seen problems in **complex algebra** are listed below for **information only.**

A collection of **two** or **more equations** in two or more variables that contain **at least one nonlinear equation** is referred to as a **system of non-linear equations.** Equations that can’t be expressed in this way are known as **non-linear equations.**

**(a)** Solving a system of **non-linear parabolic equations and linear equations** to find their **intersection point** or points**(b)** Solving a system of** non-linear circle equations** and **linear equations** to find their **intersection point** or points**(c)** Solving a system of** non-linear circle equations** and **non-linear eclipse equations** to find their **intersection point** or points

While **solving** such systems, we normally use the **substitution strategy** that we use for linear systems. After the **first equation** is resolved for a **single variable,** the **second equation** is resolved for a **second variable** using the **first solution,** and so on. However, the problem is still too complex to cover here.

## Numerical Example

Solve the following **non-linear quadratic equation** for the values of x:

x^{2} + 10x +1 = 0

### Solution

**Comparing** the given equation with the **standard quadratic equation:**

**a = 1, b = 10, c = 1**

Using the following **formula** for the solution of **quadratic equations:**

\[ x = \dfrac{ – b \pm \sqrt{ b^2 – 4ac } }{ 2a } \]

**Substituting values:**

\[ x = \dfrac{ – (10) \pm \sqrt{ (10)^2 – 4(1)(1) } }{ 2(1) } \]

\[ x = \dfrac{ – 10 \pm \sqrt{ 96 } }{ 2 } \]

\[ x = \dfrac{ – 10 \pm 9.8 }{ 2 } \]

\[ x = \dfrac{ – 19.8 }{ 2 }, \dfrac{ -0.2 }{ 2 } \]

**x = = -9.9, -0.1**

*All images were created with GeoGebra.*