Question

Partial differential equation p^2 + q2 = 16

Answer 1

Answer:

A solution or integral of a partial differential equation is a relation connecting the dependent and the independent variables which satisfies the given differential equation. A partial differential equation can result both from elimination of arbitrary constants and from elimination of arbitrary functions as explained in section 1.2. But, there is a basic difference in the two forms of solutions. A solution containing as many arbitrary constants as there are independent variables is called a complete integral. Here, the partial differential equations contain only two independent variables so that the complete integral will include two constants.A solution obtained by giving particular values to the arbitrary constants in a complete integral is called a particular integral.

Singular Integral

Let f (x,y,z,p,q) = 0 ---------- (1)

be the partial differential equation whose complete integral is

f (x,y,z,a,b) = 0----------- (2)

where „a‟ and „b‟ are arbitrary constants.

Differentiating (2) partially w.r.t. a and b, we obtain

The eliminant of „a‟ and „b‟ from the equations (2), (3) and (4), when it exists, is called the singular integral of (1).

General Integral

In the complete integral (2), put b = F(a), we get

f (x,y,z,a, F(a) ) = 0 ---------- (5)

Differentiating (2), partially w.r.t.a, we get

The eliminant of „a‟ between (5) and (6), if it exists, is called the general integral of (1).

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