$y=(x+C)(\dfrac{x+2}{x-2})$

This **article aims** to find the **transient terms** from the **general solution** of the **differential equation**. In mathematics, a **differential equation** is defined as an **equation that relates one or more unknown functions and their derivatives**. In applications, functions generally represent physical quantities, **derivatives** represent their **rates of change**, and a differential equation defines the relationship between them. Such relationships are common; therefore, **differential equations** are essential in many disciplines, including **engineering**,** physics**, **economics,** and** biology.**

**Example**

In **classical mechanics**, the **movement of a body** is described by its **position** and **velocity** as the **time value changes.** **Newton’s laws** help these variables to be expressed dynamically (given **position**, **velocity**, **acceleration**, and **various forces acting on the body**) as a differential equation for the unknown position of the body as a function of time. In some cases, this **differential equation** (called the equation of motion) can be solved explicitly.

**Types of differential equations**

There are** three main types** of differential equations.

**Ordinary**differential equations**Partial**differential equations**Non-linear**differential equations

** **

**Ordinary differential equations**

An **ordinary differential equation** (ODE) is an **equation** containing an unknown function of **one real or complex variable** $y$, its derivatives, and some given function of $x$. The **unknown function** is represented by a variable (often denoted $y$), which therefore depends on $x$. Therefore, $x$ is often called the independent variable of the equation. The term “ordinary” is used in contrast to the **partial differential equation,** which may concern more than one **independent variable.**

**Partial** **differential equations**

A **partial differential equation** (PDE) is an equation that contains unknown functions of **multiple variables** and their **partial derivatives.** (This contrasts **ordinary differential equations,** which deal with parts of one variable and its derivatives.) **PDEs** formulate problems involving functions of several variables and are either solved in closed form or used to create the appropriate computer.

**Non-linear differential equations**

A **non-linear differential equation** is an equation that is not linear in the **unknown function and its derivatives** (linearity or nonlinearity in the arguments of the function is not considered here). There are very **few methods for solving non-linear differential equations** exactly; known ones typically depend on an equation with particular symmetries. **Non-linear differential equations** exhibit **highly complex behavior** in extended time intervals, characteristic of chaos.

**Expert Answer**

**By solving the given equation:**

\[y=(x+C)(\dfrac{x+2}{x-2})\]

\[(x+C)(\dfrac{x+2}{x-2})=\dfrac{x^{2}}{x-2}+\dfrac{(2+C)x}{x-2}+\dfrac{2C}{x-2}\]

Take the **limits of each of three terms** to $x\rightarrow\infty$ and observe which **t****erms approaches zero.**

All the **three terms are rational expressions**, so the term $\dfrac{2C}{x-2}$ is a **transient term.**

**Numerical Result**

**The term** $\dfrac{2C}{x-2}$ is a **transient term.**

**Example**

Find the transient terms in this general solution of the differential equation, if any.

$z=(y+C)(\dfrac{y+2}{y-2})$

**Solution**

**By solving the given equation:**

\[z=(y+C)(\dfrac{y+4}{y-4})\]

\[(y+C)(\dfrac{y+4}{y-4})=\dfrac{y^{2}}{y-4}+\dfrac{(2+C)y}{y-2}+\dfrac{2C}{y-2}\]

Take the **limits of each of three terms** to $x\rightarrow\infty$ and observe which t**erms approaches zero.**

All the **three terms are rational expressions**, so the term $\dfrac{2C}{y-2}$ is a **transient term.**

**The term** $\dfrac{2C}{y-2}$ is a **transient term.**