Monday, 16 December 2013

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   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  .

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



No comments:

Post a Comment

Our Latest Post

Introduction of Circle

All of my lessons and teaching videos are in English and most of them are for students of Logistics Management. But many of mys students an...

Popular Post