Page 266 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 17: Lagrange’s Methods for Solving Partial Differential Equations

Put the values of a and b, the singular integral is Notes

x 2 y 2 c 2
z = 2 2 2 2 2 2 2 2 2 ,
(c x y ) (c x y ) (c x y )
2
2 2
2
2
2
2
2
or z (c x y ) = (c x y )
2
2
2
or x + y + z 2 = c .
Self Assessment
Find a complete integral of following equations:
11. z = px + qy + pq.
2
2
12. z = px + qy + p + q .
2
2
13. z = px + qy + ( p + q + ).
Standard III
The equations which do not contain x and y, i.e., which are of the form
F (z, p, q) = 0 ... (1)

can be solved in the following way.
, ,
Write x + ay = X where a is an arbitrary constant and assume z to be a function of (x + ay) i.e. of
X alone.

z = f (X) when X = (x + ay);
z dz X dz
p = .1,
x dX x dX

z dz X dz
q = . . a .
y dX y dX
Now the equation (1) becomes
dz dx
F , z , a 0
dX dX

which is an ordinary differential equation of the first order and can be integrated. So the complete
integral will be known.
The general and singular integrals can be found as in first two cases.

Illustrative Examples

2
2
Example 1: Find a complete integral of: 9(p z + q ) = 4.
Solution:
Put z = f (x + ay) = f (X)

z dz X dz
p = .
x dX x dX

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