Partial Differential Equations

The Equations which contain partial differential coefficients , independent variables and dependent variables are known as Partial Differential Equations .

If independent variables are denoted by x & y and dependent variable denoted by z, then partial differential coefficients are denoted as :

$\displaystyle \frac{\partial z}{\partial x} = p $ , $\displaystyle \frac{\partial z}{\partial y} = q $

$\displaystyle \frac{\partial^2 z}{\partial x^2} = r $ , $\displaystyle \frac{\partial^2 z}{\partial x \; \partial y} = s $ , $\displaystyle\frac{\partial^2 z}{\partial y^2} = t $

Example: Form the partial differential equation from z = f(x2 – y2)

Solution: z = f(x2 – y2)

Differentiating withe respect to x and y ,

$\displaystyle p = \frac{\partial z}{\partial x} = f'(x^2 – y^2) 2 x $

$\displaystyle q = \frac{\partial z}{\partial y} = f'(x^2 – y^2) (- 2 y ) $

On dividing we get ,

$\displaystyle \frac{p}{q} = \frac{-x}{y} $

$\displaystyle p y = – q x $

$\displaystyle p y + q x = 0 $

Lagrange’s Linear Equation & The Auxiliary Equations

If P p + Q q = R ; Where P , Q , R are the functions of x , y , z and $\displaystyle \frac{\partial z}{\partial x} = p $ , $\displaystyle \frac{\partial z}{\partial y} = q $

The Auxiliary Equations are :

$\displaystyle \frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R} $

Example: Solve the partial differential equations : y2 p – x y q = x (z – 2 y)

Solution : $\displaystyle y^2 p – x y q = x (z – 2 y)$

The Auxiliary Equations are :

$\displaystyle \frac{dx}{y^2} = \frac{dy}{- x y} = \frac{dz}{x (z – 2 y)} $

By considering first two ,

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

$\displaystyle x dx = – y dy $

On Integrating ,

$\displaystyle \int x dx = – \int y dy $

$\displaystyle \frac{x^2}{2} = – \frac{y^2}{2} + \frac{C_1}{2} $

$\displaystyle x^2 + y^2 = C_1$ …(i)

By considering last two Equations ,

$\displaystyle \frac{dy}{- y} = \frac{dz}{(z – 2 y)} $

$\displaystyle – z dy + 2 y dy = y dz $

$\displaystyle 2 y dy = y dz + z dy $

On Integrating ,

$\displaystyle 2 ( \frac{y^2}{2} ) = y z + C_2 $

$\displaystyle y^2 = y z + C_2 $ …(ii)