Introduction to differential equation

Introduction:

Equation which contains derivative is called differential equation . Example;

$$ \frac{dy}{dx}=\frac{-y}{x}$$

Is ordinary differential equation because it contains full derivatives.

Order of the equation:

Order is represented by how much that it is differentiated. i.e.

\(\frac{dy}{dx}= \frac{-y}{x} \) has order 1.

Power of differential equation:

Power of the differential equaiton is represented by maximum power of highest order derivative. i.e.

$$\biggl(\frac{dy}{dx}\biggr)^2=\frac{-y}{x}$$

Order= 1

Power = 2

Example;

$$\sqrt{\frac{dy}{dx}}+3x\frac{dy}{dx}+5=0$$

$$or,\;\;\;\biggl(3x\cdot\frac{dy}{dx}+ 5\biggr)^2= \biggl(-\sqrt{\frac{dy}{dx}}\biggr)^2$$

$$or,\;\;\; 9x^2\biggl(\frac{dy}{dx}\biggr)^2+ 30 x\cdot\frac{dy}{dx}+ 25= \frac{dy}{dx}$$

order= 1

power=2

Standard form of differential equation:

$$\frac{d^2y}{dx^2}+ p(x)\frac{dy}{dx}+ Q(x) y= R(x)$$

Linear, second order, non- homogeneous ordinary differential equation.

For homogeneous equation:

$$\frac{d^2y}{dx^2}+ p(x) \frac{dy}{dx}+ Q(x) y=0$$

If \(p(x_\circ\)) and \(Q(x_\circ\)) are analytic then the differential equation has solution.

Note:

Check: \(\lim\limits_{x\to x_\circ} x p(x)= finite\)

\(\lim\limits_{x\to x_\circ} (x-x_\circ)^2Q(x)= finite\)

Bassel equion,

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

$$\frac{d^2y}{dx^2}+\frac1x \frac{dy}{dx}+\biggl(1-\frac{n^2}{x^2}\biggr)y=0$$

$$p(x)=\frac1x, \;\;\;\; Q(x)=\biggl(1-\frac{n^2}{x^2}\biggr)$$

$$\lim\limits_{x\to 0} p(x)\to\infty\;\;\;, Q(x)=\biggl(1-\frac{n^2}{x^2}\biggr)$$

NOw,

$$\lim\limits_{x\to 0} x p(x)= \lim\limits_{ x\to 0 }x\cdot\frac1x=1 (finite)$$

$$\lim\limits_{x\to 0} (x-0)^2 Q(x)= \lim\limits_{x\to 0} x^2\cdot \biggl( \frac{x^2-n^2}{x^2}\biggr)= - n^2 ( finite)$$

\(\therefore\) At least one solution exits.

Exact differential equation:

$$ F( x, y) = c$$

$$ dx= \frac{\delta F}{\delta x}dx+\frac{\delta F}{\delta y} dy=0$$

$$or,\;\;\; \frac{\delta F}{\delta x}dx+\frac{\delta F}{\delta y} dy=0$$

$$or,\;\;\; M(x,y) dx+ N(x,y) dy=0$$

$$or,\;\;\; \frac{\delta M}{\delta y}=\frac{\delta N}{\delta x}$$

This is exact differential equation.

$$or,\;\;\; \frac{\delta ^2 F}{\delta x\delta y}=\frac{\delta ^2 F}{\delta x\delta y}$$

This is perfect differential quantity ( independent of path).

IN thermodynamics the quantity which are perfect differential represent the state of system. Eg, ( P, V, T ,S)

Wronskian:

Suppose a second order differential equation

$$ y''+ p(x) y'+ q(x) y=0$$

Which has two solution. Let \(y_1\) and \(y_2\) be it's solution.

Then, we define determine of \(y_1\) and \(y_2\) as,

$$\begin{vmatrix} Y_1 & Y_2\\ Y_1' & Y_2'\\ \end{vmatrix}$$

To check dependent or independent,

if w=0 ( independent)

w\(\ne 0\) ( dependent)

Application of differential equation in physics:

From Mechanics,

$$ F_y= - ky$$

$$ or\;\; m\frac{d^2y}{dt^2}+ ky=0 $$

$$or,\;\;\; \frac{d^2y}{dt^2}+ \frac{k}{m}y=0$$

$$or,\;\;\; \frac{d^2y}{dt^2}+ \omega^2 y=0$$

where,

$$ \omega^2=\frac km$$

$$or,\;\;\; \omega=\sqrt{\frac km}$$

$$\Rightarrow f= \frac{1}{2\pi}\sqrt{\frac km}$$

Let, \(\frac{d}{dt}=D\)

$$or,\;\;\; ( D^2+ \omega^2) y=0$$

$$or,\;\;\; (D+ \omega)( D- \omega ) y=0$$

Since second order differential equation has two solution.

So,

$$( D+ i\omega )yy_1=0 \;or,\; (D - i\omega) y_2=0$$

$$or, \frac{dy_1}{y_1}= -i\omega dt$$

On integrating

$$ y_1= Ae^{-i\omega t} \; and \; y_2= B e^{+i\omega t}$$

The complete solution of above differential equation

$$y= y_1 + y_2$$

$$= Ae^{-i\omega t}+ Be^{+i\omega t}$$

Which is according to superposition principle.

Reference:

1. Dass, H. K.Mathematical Physics. New Delhi: S. Chand, 2005.
2. Vaughn, Michael T.Introduction to Mathematical Physics. Weinheim: Wiley-VCH, 2007
3. Butkov, Eugene.Mathematical Physics. Reading, MA: Addison-Wesley Pub., 1968.
4. Carroll, Robert W.Mathematical Physics. Amsterdam: North-Holland, 1988.
5. Adhikari, Pitri Bhakta, and Dya Nidhi Chhatkuli.A Textbook of Physics. Third Revised Edition ed. Vol. III. Kathmandu: SUKUNDA PUSTAK BHAWAN, 2072.