Page 355 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes We can have various types of partial differential equations.
1. Linear partial differential equations with constant coefficients
We may have equations of the type
C r + C s + C t + C p + C q + C z = f(x, y)
1 2 3 4 5 6
where C , C , C , C , C are constants. We have already given the methods of solving these
1 2 3 4 5
types of equations in the earlier unit no. 20.
2 2
z z
x
y
The examples are 2 2 f ( , )
x y
2
z
x
y
f ( , )
x y
2 2
z 1 z
x 2 C 2 y 2
2 z z
K 2 (here K is a constant)
x y
2. Equations with Variable Coefficients
In this type of partial differential equations we will have a structure as follows
Rr + Ss + Tt + f(x, y, z, p, q) = 0 ...(1a)
where R, S, T are functions of x, y, z.
As suggested in the section (21.1) we classify this equation into three classes
2
(a) Hyperbolic if s 4rt > 0
2
(b) Parabolic if s 4rt = 0 and
2
(c) Elliptic if s 4rt < 0
In dealing with equations of the above types first we reduce them to canonical form. The
solution of Laplace equation, Wave equation and conduction of heat or diffusion we defer
cases to next two units.
3. Equations reducible to homogeneous linear form
An equation in which the coefficient of a differential coefficient of any order is a constant
multiple of the variables of the same degree, may be transformed into one having constant
coefficients.
Example: Transform the equation

2 z 2 z z z
x 2 2 y 2 2 y x 0 ...(1)
x y y x
into a form with constant coefficients.
Solution: Put u = log x, v = log y
z z 1
.
x u x
z z
or x
x u

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