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Partial Differential Equations

There are many applications of partial differences. This equation is complicated because we would be dealing with more than one independent variable. You will find three parts variable that are independent in a particular partial difference variables. The coefficient denoted as $\frac{partial\;z}{partial\;x}=p$ and $\frac{partial\;z}{partial\;y}=q$ and dependent variable is z. The classic ways where Partial Difference Equations are used are acoustics, heat transfer, electrodynamics and fluid flow.

Another way to write the partial difference is $F\left(x,y,f_{x},f_{y},f_{xy},f_{yz}…..\right)=0$.

Partial equation solving rule:

  1. Write auxillary equations is, $\frac{dx}{P}$ = $\frac{dy}{Q}$ = $\frac{dz}{R}$
  2. Let the two solution $u = c_{1}$, $v = c_{2}$ and solve the auxillary equation.
  3. Find the solution $f(u, v)$ = 0.

Solved Examples

Question 1: Find the solution of partial differential equation $x^{3}p + y^{3}q = z^{3}$.

Solution:

Given: $x^{3}p + y^{3}q = z^{3}$

Step 1: Auxillary equation

$\frac{-2}{x}$ = $\frac{-2}{y}$ $+ C$……(1)

and

$\frac{-2}{y}$ = $\frac{-2}{z}$ $+ C$ ….(2)

Step 2: From equation (1) and (2) eliminate C

$\frac{-2}{x}$ = $\frac{-2}{y}$ + $\frac{2}{z}$$\frac{2}{y}$

$\frac{2}{z}$ = $\frac{4}{y}$$\frac{2}{x}$

Step 3: By simplifying the equation

$2(2x – y) = xy$

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