Page 245 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Differentiate (1) with respect to x we have
p = a ...(3)
Also differentiate (1) with respect to y we have

z
= q = b ...(4)
y
Substituting the values of a and b from (3) and (4) into the equation (1) we have
2
z = px + qy + p + q 2 ...(2)
so equation (1) having two arbitrary constants a and b is the complete integral of partial
differential equation (2).
Differentiating (1) with respect to a and b respectively,

we get
0 x 2a
and ...(5)
0 y 2b
Substituting the values of a and b in (1) we have

x 2 y 2 x 2 y 2
Z =
2 2 4 4
2
2
4Z = (x + y ) ...(6)
To see whether equation (6) satisfies (2) we have
4p 2x
4q 2y

Substituting in R.H.S. of (2) we have
x 2 y 2 x 2 y 2 (x 2 y 2 )
= z L.H.S.
2 2 4 4 4
So equation (6) satisfies equation (2).

Equation (6) represents a paraboloid of revolution, the envelops of all the planes represented by
the complete integral. Equation (6) represents singular integral.

Example 2: Show that
2
Z = be ax + a y ...(1)
is the complete integral of partial differential equation

p 2 = zy ...(2)
Differentiating (1) w.r.t. x, y respectively

z ax+a y
2
p = bae ...(3)
x
z 2
2
q = ba e ax+a y ...(4)
y

238 LOVELY PROFESSIONAL UNIVERSITY