Page 271 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
Z Z z
. = zp = P (say)
x z x
Z Z z
. = zq = Q (say)
y z y

The given equation becomes
2
2
2
2
P + Q = x + y .
2
2
2
2
P x = y Q .
2
2
2
2
Let P x = y Q = a 2
2
2
2
2
or P = (a + x ) and Q = (y a ).
2
2
2
2
dZ = P dx + Q dy = (x + a ) dx + (y a ) dy
x 2 2 a 2 2 2 y 2 2 a 2 2 2
Z = (x a ) log[x (x a )] (y a ) log[y (y a )] c .
2 2 2 2
Complete integral is
2
2
2
2
2
2
2
2
2
2
2
z = x (x + a ) + a log [x + (x + a )] + y (y a ) a log [y + (y a )] + k.
2
2
2
2
Example 3: Solve: (x + y ) (p + q ) = 1.
Solution:
Put x = r cos , y = r sin ,
y
2
2
2
i.e. r = x + y , = tan 1 .
x
z z r z z sin z
p . . cos . ,
x r x x r r
z z r z z cos z
q . . sin . .
y r y y r r
On substitution the equation becomes
2 2
z 1 z
2
r 1
r r 2
2 2
z z
2
or r 1
r
which is of the form f (q, x) = f (p, y).
1 2
Putting
2 2
z 2 z
2
r a 1 ,
r
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