Equations of First Order

 

In this post we will discuss about the equation of first order but not first degree .It is usually denoted
      
                dy/dx    by   p .

Thee are three types of such equations

1) Equations solvable for  p  .
2)Equation solvable for    y  .
3)Equation solvable for    x  .

Equation Solvable for   p   :- 

examples like this    p.p +2py  cot x = y.y   and its solution is

[y-(c/1+cos x)][y-(c/1-cos x)] = 0

is the equation solvable for   p   .

Equation Solvable for    y   :-

Let in the given differential equation , on solving for   y   , given that ;

                    y=f(x,p)  ---------------(1)

Differentiating with respect to   x   , we obtain ;

              p=dy/dx=A(x,p,dp/dx)

so that we obtain a new differential equation with variables    x   and   p    .

Suppose that it is possible to solve the equation

Let the solution be
                            F(x,p,c)=0  ----------(2)
                          where    c   is the arbitrary constant .

The equation of  (1)  may be exhibited in either of the two forms . We may either eliminate   p   between  (1)  and  (2)  and obtain  A(x,y,c)   as the required solution or we may solve   (1)  and  (2)  for  x , y   and obtain .

                   x=f'(p,c)   and   y=f"(p,c)

as required solution where   p  is the parameter .

Equations Solvable for    x   :-

Let the given differential equation , on solving for  x   , gives

                                 x=f(p,y)  -----------------------(1)

differentiating with respect to   y  we obtain

                           1/p=dy/dx=A(y,p,dp/dx)  ; say

So that we obtain a new differential equation in variables    y      and   p   , Suppose that it is possible to solve the equation .

Let the solution be
                              F(p,y,c)=0   -------------------(2)

After the elimination    p   between    (1)   and   (2)     will give the solution .  Express   x  and   y   in terms    of   p   and   c   where    p   is to be regarded as parameter .