A **differential equation** is an equation which involves the derivatives or differentials with or without the dependent variable or independent variable or both. A differential equation is said to be ordinary if there is no partial derivative. Consider a real valued function of a real variable $(ƒ:R\to R)$ in the form \[y=ƒ(x)\]

If this function is differentiable at a point $x=c\in R$, we write \[\left.\frac{dy}{dx}\right|_{x=c}=ƒ'(c)\]

If it is differentiable for all values of $x\in R$, we write \[\frac{dy}{dx}=ƒ'(x)=g(x)\text{ (say)}\]

In particular, if we have a simple function defined by \[y=\frac{c}{x}\text{ or }xy=c\] where $c$ is a constant, simple differentiation of this function gives \[\frac{dy}{dx}=\frac{-c}{x^2}\text{ __(1)}\]

This is an equation which involves the first derivative of the function defined by $y=\frac{c}{x}$, the independent variable $x$ and the constant $-c$. The equation $\text{(1)}$ can be put and classified into following different standard forms:

### Variables Separated Form (Standard Form I)

\[dy=\frac{-c}{x^2}dx\] in which the variables $x$ and $y$ are separated. Hence, variable separated form is an equation of the form \[Ydx=Xdx\] where $Y$ is a function of $y$ alone and $X$ is a function of $x$ alone.

### Homogeneous Form (Standard Form II)

\[\frac{dy}{dx}=-\left(\frac{c/x}{x}\right)=-\frac{y}{x}\] in which $x$ and $y$ appear in the form $\frac{y}{x}$ only. Hence, homogeneous form is an equation of the form \[\frac{dy}{dx}=ƒ\left(\frac{y}{x}\right)\] where $ƒ\left(\frac{y}{x}\right)$ is a function of $\frac{y}{x}$.

### Linear Form (Standard Form III)

\[\frac{dy}{dx}=\frac{1}{x}y=0\] in which the coefficient of $y$ is a function of $x$ alone. Hence, linear form is an equation of the form \[\frac{dy}{dx}+Py=Q\] where $P$ and $Q$ are the functions of $x$ only.

### Exact Form (Standard Form IV)

\[ydx+xdy=0\] in which the left hand side could be written as a single (exact) differential $d(xy)$. Hence, exact form is an equation of the form \[M(x,y)dx+N(x,y)dy=0\] where $M$ and $N$ are the functions of $x$ and $y$ such that the left hand side of this equation can be expressed as a single (or perfect or exact) differential of the form $dƒ(x,y)$, where $ƒ(x,y)$ is a function of $x$ and $y$.

Thus, we observe that an equation may contain the differential coefficient $\left(\frac{dy}{dx}\right)$, or differentials $dx$ and $dy$, with or without the dependent variable $y$ or independent variable $x$ or both.

In each of the examples mentioned above, the highest derivative of $y$ is one. An equation like this is called an **ordinary differential equation of the first order**. We also notice that the highest power of the highest (here one) derivative is also one. Such equation is known as an **ordinary** (having only one independent variable $x$) **differential equation of the first order and first degree**.

A differential equation is said to be of **order one** if the order of the highest derivative is one. It is said to be of order $n$ if the order of the highest derivative appearing in it is $n$. The differential equations $\frac{dy}{dx}=-\frac{y}{x}$ and $\frac{d^2y}{dx^2}+y=\sin x$ are of order one and two respectively.

## Degree and Solution of a Differential Equation

The **degree** of a differential equation is the power to which the highest derivative in it is raised. The degree of the differential equations $\left(\frac{dy}{dx}\right)^2=\left(\frac{y}{x}\right)^2$ and $\frac{d^2y}{dx^2}+y=x^n$ are two and one respectively.

A **solution** of a differential equation is any relation between the variables, which is free from derivatives or differentials and which satisfy the equation identically. The equation \[\frac{dy}{dx}=-\frac{y}{x}\text{ __(i)}\] is satisfied by \[y=\frac{1}{x}\text{ __(ii)}\]

So, it is a solution of $\text{(i)}$. It is also satisfied by \[y=\frac{c}{x}\text{ __(iii)}\] where $c$ is an arbitrary constant.

The solution $\text{(ii)}$ of $\text{(i)}$ is called a **particular solution** or **particular integral** (P.I.) of $\text{(i)}$. The solution $\text{(ii)}$ is obtained by substituting $y=1$ when $x=1$, hence it is known as particular solution. The solution $\text{(iii)}$ of $\text{(i)}$ containing one arbitrary constant is called the **general solution** of the differential equation of order one. Therefore, a general solution of a differential equation of order $n$ is defined as a solution which contains $n$ arbitrary constants. A general solution is also known as a **complete solution** or **complete primitive**.

## Equations of the First Order and First Degree

An ordinary differential equation of the first order and first degree may be written in the form \[\frac{dy}{dx}=ƒ(x,y)\] where $ƒ(x,y)$ is a function of $x$ and $y$. If \[ƒ(x,y)=-\frac{M(x,y)}{N(x,y)}\] where $M$ and $N$ are the functions of $x$ and $y$, we may write it in the form \[M(x,y)dx+N(x,y)dy=0\]

## Autonomous Differential Equation

If an equation could be written in the form \[\frac{dy}{dx}=ƒ(y)\text{ or, }y’=ƒ(y)\] where $ƒ(y)$ is a function of $y$ alone, then the equation is called an **autonomous differential equation**. The equation \[\frac{dy}{dx}=y\] is autonomous.

**Next:** Variables Separated Form (Standard Form I)

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