Page 264 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

which is of the form F (p, q) = 0, [Standard I] Notes
Solution is z + aX + bY + c
2
where a = b .
2
z = b (x + y) + bxy + c.
Self Assessment

Find the complete integrals of:

2
2
2
7. p + q = m .
8. pq = k.
2
2
9. p + q = npq.
10. p q 1.

Standard II

The equation
z = px + qy + f (p, q),
which is analogous to Clairaut’s form, has for its complete integral
z = ax + by + f (a, b) ... (1)

z z
for = p = a and = q = b
x y
In order to obtain the general integral put b = (a).

z = ax + y (a) + f {a, (a)}.
Differentiating with respect to a,
0 = x + y (a) + f (a)
and eliminate a between these equations.

In order to obtain the singular integral, differentiate (1) with respect to a and b, i.e.,
0 = x + f/ a, ... (2)
0 = y + f/ b ... (3)
and eliminate a and b between the equations (1), (2) and (3).

Illustrative Examples

Example 1: Solve z = px + qy 2 (pq).
Solution:
The complete integral is

z = ax + by 2 (ab) ... (1)

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