Page 223 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 2 n
dy d y d y
y
F x , , , 2 ,..., n = 0 ...(1)
dx dx dx
involving up to nth derivative of y.
In contrast to the above we may sometimes have to deal with a dependent variable and more
than one independent variables. Thus we may have partial derivatives of the dependent variable
u with respect to independent variable x, y, z,.... So we have partial derivatives of u in the
u u u
differential equation like , , etc. We may have a higher partial derivatives also present
x y z
2 u 2 u 2 u
in the differential equations i.e. , , ,.... . Such a differential equations involving one
x 2 x y z 2
dependent variable u and a number of independent variables x, y, z, ... along with the partial
derivatives of u with respect to x, y, z, .. is known as partial differential equation i.e.

u u u 2 u
y
u
z
f x , , , ... , , , ,... ,.... = 0 ...(2)
x y z x 2
We may have a situation in which the partial differential equation involves only first derivatives
only. Such an equation is known as first order partial differential equation i.e.
u u
z
y
f x , , ,... , ,... = 0 ...(3)
u
1
x x
n
Here the order of the equation is one and it is known as first order partial differential equation.
Let us denote independent variables, as x, y and z as dependent variable. Also let us put
z
p =
x
...(4)
z
q =
y
z z
So the partial differential equation involving x, y, z, , will be of the form
x y
p
q
y
x
f 2 ( , , , , ) = 0 ...(4)
z
Example: The equation
2 z
z
x 2 = y
is a partial differential equation of second order. The equation

2
z z
= 0
x y
is a first order partial differential equation and of second degree involving two independent
variables x and y. The equation

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