Page 249 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes is a complete integral of the partial differential equation
Z = px + qy + p q pq ...(8)
Also find the singular integral.

Solution: Differentiate (7) with respect to a and b respectively, i.e.,
0 = x + 1 b ...(9)

0 = y 1 a ...(10)
So a = y 1, b = x + 1
Substituting values of a and b in (7) we have

z = x(y 1) + y(x + 1) + y 1 x 1 (y 1)(x + 1)
Simplifying, we have

z = xy x + y 1
as singular integral. Differentiating (7) with respect to x and y separately we have
Z Z
p , a q , b substituting in (7)
x y
we have

z = px + qy + p q pq
which is just equation (8). So (7) is the complete integral of (8).

Self Assessment

6. Find the singular integral for the differential equation
Z = px + qy + p/q

16.5 Summary

 The partial differential equation of the first order can be a function of x, y, z and the partial
z z
derivatives of z i.e., p and . q
x y
 The differential equation can have a solution depending upon two unknown constants.
Such a solution is called complete integral.

 If we substitute some fixed values for the constants we get particular integral.
 On the other hand if we get the solution of the equation in the form
(u, v) = 0
where u, v are known functions of x, y, z then we get a general solution.
16.6 Keyword

By varying the two arbitrary constants we can get various integrals or solutions of the partial
differential equations. It is advisable to visualize geometrically the integral surfaces or integral
curves.

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