An equation of the form

**y = px + f (p) , where p = dy/dx**

is called Clairaut's Equation .

Differentiating both sides of the equation with respect to

**x**, we have ,

**p = p + xdp/dx + f' (p) dp/dx or dp/dx {x + f' (p)} = 0**

therefore , either ,

**dp/dx = 0 ------------ (1)**

or ,

**x + f' (p) =0 ------------(2)**

from (1) ,

**p = C --------------(3)**

Now if

**p**be eliminated between

**(3)**and the original equation , we get

**y = Cx + f(C)**as the general or complete solution of the equation .

Again , if

**p**be eliminated between

**(2)**an the original equation , we shall obtain a relation between

**x and y**which also satisfies the differential equation , and as such can be called a solution of the given equation . Since this solution does not contain any arbitrary constant nor can it be derived from the complete solution by giving any particular value to the arbitrary constant , it is called the Singular Solution of the differential equation .

**Thus we see that the Equation of Clairaut's form has two kinds of solution .**

a) The complete solution ( linear in

**x**and

**y**) containing one arbitrary constant .

b) The singular solution containing no arbitrary constant .

Now , to eliminate

**p**between

**y = px + f(p) and 0 = x + f'(x)**

is the same as to eliminate

**C**between ,

**y = Cx + f(C) and 0 = x + f'(C)**

i.e. , the same as the process of finding the envelop of the line

**y = Cx + f(C)**for different values of

**C**.

Thus , the singular solution represents the envelope of the family of straight lines represented by the complete solution .

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