Page 261 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Standard I

In this form of the equation only p and q are present. The partial differential equation will be of
the form
f (p, q) = 0 ... (1)

in which x, y, z do not appear. The complete integral is
z = ax + by + c ... (2)
where a and b are connected by the relation
f (a, b) = 0 ... (3)

z z
Since p a and q b , which on substitution becomes the given equation (1).
x y
To find the general solution, let from (3) put b = (a) and replacing c by (a), we have

z = ax + (a)y + (a) ... (4)
Differentiating (4) with respect to a,
0 = x + y (a) + (a) ... (5)

The general solution is obtained by eliminating a between (4) and (5).
Suppose from (2), b = (a) and replacing c by (a) the general solution is obtained by eliminating
‘a’ between the following equations:

z = ax + (a) y + (a). ... (6)
Differentiating (3) with respect to a,
0 = x + y (a) + (a) ... (7)
Now to find the singular integral, differentiate
z = ax + (a) y + c

with respect to a and c,
0 = x + y (a)
and 0 = 1.

Now the last equation shows that there is no singular integral.
Illustrative Examples

Example 1: Solve: q = exp. ( p/a).
Solution:

The complete integral is
z = x + y +
where = exp. ( /a)
i.e., the complete integral is
z = x + {exp. ( /a} y +

254 LOVELY PROFESSIONAL UNIVERSITY