Introduction to Differential Equation

Basic Definition

An equation containing an independent variable, dependent variable and differential coefficients of dependent variable with respect to independent variable is called a differential equation. An ordinary differential equation is one is which there is only one independent variable.

For Example :

$x\frac{dy}{dx}=3y$ .........(1)

$x\frac{d^{2}y}{d{x}^2}-6\frac{dy}{dx}+3y=0$ .........(2)

$\sqrt{\frac{dy}{dx}}+y=x$ ..........(3)

${\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^{\frac{3}{2}}={\left(\frac{d^{3}y}{d{x}^3}\right)}^2$ .........(4)

${\left(\frac{dy}{dx}\right)}^2+{\left(\frac{d^{2}y}{d{x}^2}\right)}^3=x$ .........(5)

$\sqrt{1+{\left(\frac{d^{2}y}{d{x}^2}\right)}^2}+{\left(\frac{dy}{dx}\right)}^3=x^2$ .........(6)

Order and Degree of a Differential Equation:

The order of differential equation is the order of highest order derivative appearing in the equation.

For Example :

Orders of differential equations (1), (2), (3), (4), (5) and (6) are 1, 2, 1, 3, 2 and 2 respectively.

The degree of differential equation is the degree of the highest order derivative involved in it, when the differential coefficients are free from radicals and fractions (i.e. write differential equations as polynomial in derivatives)

For Example :

Degrees of differential equations (1), (2), (3), (4), (5) and (6) are 1, 1, 1, 4, 3 and 2 respectively.

Illustration 1: Find the order and degree (if defined) of the following differential equations :

(i) ${\left(\frac{d^{3}y}{d{x}^3}\right)}^{\frac{2}{3}}=\frac{dy}{dx}+2$

(ii) $\frac{d^{2}y}{d{x}^2}=xIn\left(\frac{dy}{dx}\right)$

Solution : (i) The given differential equation can be re-written as

${\left(\frac{d^{3}y}{d{x}^3}\right)}^2={\left(\frac{dy}{dx}+2\right)}^3$. Hence its order is 3 and degree 2.

(ii) $\frac{d^{2}y}{d{x}^2}=x\left(\frac{dy}{dx}\right)$. Hence its order is 2 and degree 1.

Formation of differential Equation:

We know y² = 4ax is a parabola whose vertex is origin and axis as the x-axis . If a is a parameter, it will represent a family of parabola with the vertex at (0, 0) and axis as y = 0 .

Differentiatingy² = 4ax . . (1)

$2y\frac{dy}{dx}=4a$ . . (2)

From (1) and (2),y² = 2yx$\frac{dy}{dx}$ $\Rightarrow$ y = 2x$\frac{dy}{dx}$

This is a differential equation for all the members of the family and it does not contain any parameter ( arbitrary constant).

(i) The differential equation of a family of curves of one parameter is a differential equation of the first order, obtained by eliminating the parameter by differentiation.

(ii) The differential equation of a family of curves of two parameter is a differential equation of the second order, obtained by eliminating the parameter by differentiating the algebraic equation twice. Similar procedure is used to find differential equation of a family of curves of three or more parameter.

Illustration 2: Find the differential equation of the family of curves y = Ae^x + Be^3x for different values of A and B.

Solution: y = A e^x + Be^3x . . . . (1)

y₁ = Ae^x + 3Be^3x . . . (2) $\left(y_1=\frac{dy}{dx}\right)$

y₂ = Ae^x + 9B^3x . . . (3) $\left(y_2=\frac{d^{2}y}{d{x}^2}\right)$

Eliminating A and B from the above three, we get

$\left|\begin{array}{c}{e}^x{e}^{3x}-y\\ e^{x}3e^{3x}-y_1\\ e^{x}9e^{3x}-y_2\end{array}\right|$ = 0 $\Rightarrow$ e^x e^3x $\left|\begin{array}{c}11-y\\ 13-y_1\\ 19-y_2\end{array}\right|$

$\Rightarrow$ 3y + 4y_{1 –} y₂ = 0 $\Rightarrow$ 3y + 4$\frac{dy}{dx}-\frac{d^{2}y}{d{x}^2}=0$

Some Results on Tangents and Normals:

(i) The equation of the tangent at P(x, y) to the curve y= f(x) is Y – y = $\frac{dy}{dx}\left(X-x\right)$

(ii) The equation of the normal at point P(x, y) to the curve y = f(x) is

Y – y = $-\frac{1}{\left(\frac{dy}{dx}\right)}$(X – x )

(iii) The length of the tangent = CP = $y\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}$

(iv) The length of the normal = PD =$y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$

(v) The length of the cartesian subtangent = CA = $y\frac{dx}{dy}$

(vi) The length of the cartesian subnormal = AD = $y\frac{dy}{dx}$

(viii) The initial ordinate of the tangent =OB = y - x$\frac{dy}{dx}$