The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was $20000$ in $1999$ and $25000$ in the year $2004$, what will be the population of the village in $2009?$

An equation containing an independent variable, dependent variable, and differential coefficients of the dependent variable with respect to the independent variable is called a differential equation.

For Example:

(i) $\frac{dy}{dx}=2xy$

(ii) $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}+6y=x^2$

2. Order and Degree of Differential Equation:

(i) The order of a differential equation is the order of the highest order derivative appearing in the equation. The order of a differential equation is a positive integer.

(ii) The degree of a differential equation is the degree of the highest order derivative. In other words, the degree of a differential equation is the power of the highest order derivative occurring in a differential equation, when it is written as a polynomial in differential coefficients.

3. Linear and Non-linear Differential Equation:

(i) A differential equation is a linear differential equation if it is expressible in the form

P0dnydxn+P1dn-1ydxn-1+P2dn-2ydxn-2+....+Pn-1dydx+Pny=Q

where, $P_0, P_1, P_2, ..., P_{n-1}, P_n$ and $Q$ are either constants or functions of an independent variable $x$.

(ii) A differential equation will be a non-linear differential equation, if

(a) its degree is more than one.

(b) any of the differential coefficients has an exponent more than one.

(c) an exponent of the dependent variable is more than one.

(d) products containing dependent variable and its differential coefficients are present.

4. Solution of Differential Equations:

(i) The solution of a differential equation is a relation between the variables involved which satisfies the differential equation. Such a relation and the derivatives obtained therefrom when substituted in the differential equation make left-hand, and right-hand sides identically equal.

(ii) The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution of the differential equation.

(iii) The solution obtained by giving particular values to the arbitrary constants in the general solution of a differential equation is called a particular solution.

5. Equations in Variable Separable Form:

A differential equation expressible in the form $f\left(x\right) dx=g\left(y\right) dy.$

The solution of this equation is given by$\int f\left(x\right)dx=\int g\left(y\right)dy+C,$ where $C$ is a constant.

6. Equations Reducible to Variable Separable Form:

A differential equation of the form dydx=f$\left(ax+by+c\right)$ can be reduced to variable separable form by the substitution $ax+by+c=v.$

7. Homogeneous Differential Equations:

If a first-order first-degree differential equation is expressible in the form dydx=fx,ygx,y, where $f(x, y)$ and $g(x, y)$ are homogeneous functions of the same degree, then it is called a homogeneous differential equation. Such type of equations can be reduced to variable separable form by the substitution $y=vx$ or, $x=vy.$

8. Solution of Linear Differential Equations of the form $\frac{dy}{dx}+Py=Q$ and $\frac{dx}{dy}+Rx=S$:

(i) If a differential equation is expressible in the form $\frac{dy}{dx}+Py=Q$, where $P$ and $Q$ are functions of $x,$ then it is called a linear differential equation. The solution of this equation is given by $y\times I.F.=\int Q\times I\mathit{.}F. dx+C$, where $I.F.$ is called an integrating factor and is given by, $I.F.=e^{\int Pdx}$

(ii) Sometimes a linear differential equation is in the form $\frac{dx}{dy}+Rx=S$, where $R$ and $S$ are functions of $y.$ The solution of this equation is given by

$x\left(e^{\int Rdy}\right)=\int \left(S{e}^{\int Rdy}\right)dy+C$.