Page 246 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 16: Classifications of Integrals of the First Order Partial Differential Equations

Notes
2
2a y
2 2
2ax
p 2 = b a e
2
2a y
2ax
2 2
qz = b a e
Thus p 2 = qz ...(2)
So (1) is the complete integral of partial differential equation (2) since it has two arbitrary
constants.
Differentiating (2) w.r.t. p and q, we get
2p = 0 ...(5)
and z = 0 ...(6)

Eliminating p, q from (2), (5) and (6) we have
z = 0
It satisfies equation (2). So it is a singular integral. Also if we put b = 0 in (1) we get
z = 0

So z = 0 is both a singular as well as a particular solution.
Self Assessment

2
2
3. Show that F = ax + by + a + ab + b z = 0
is the complete integral of the partial differential equation
2
Z = px + qy + p + pq + q 2
and find the singular integral

4. Show that

1 2 2
F = ax + by + a b Z = 0
2
is the complete integral of the partial differential equation

1 2 2
Z = px + qy + p q
2
Find the singular integral of this partial differential equation.

16.3 General Integrals

Consider the partial differential equation of the first order
F(x, y, z, p, q) = 0 ...(1)

If on integration we get a solution of the form
f(u, v) = 0 ...(2)
where u and v are functions of x, y, z we call it a general integral. This will be illustrated by
means of the following example.

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