Clairaut's Equation

  

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 .