Ordinary Differential Equation

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.

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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