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Thursday, 19 December 2013

Homogeneous Equations

      
We have to understand about Homogeneous Equation in Calculus .

If  and  N  of the equation   Mdx + Ndy = 0   are both of the same degree in  x  and  y   , and are homogeneous , the equation is said to be homogeneous  . Such an equation can be put in the form

                                          dy/dx = f ( y/x )

Every homogeneous equation of the above type can be easily solved by putting     y = vx       
where   v  is a function of , and consequently

                                          dy/dx = v + x ( dv/dx )  

whereby it reduced to the form     v + xdv/dx = f ( v ) 

                                      i.e.        dx/x = dv/ [ f(v) - v ] 

in which the variables are separated

A Special Form :-

The equation of the form

                         dy/dx = [ ( a1x +b1y +c1 ) / ( a2x + b2y +c2 ) ] 

                         where ,  [ a1/a2 is not equal to b1/b2 ] --------------(1)

can be easily solved by putting
                                               x = x' + h 
                           and              y = y' + k    
                           where   and are constant

So that ,              dx = dx'    and      dy = dy' 
                         and choosing   h  and  k  in such a way that


         a1h + b1k + c1 = 0    and    a2h + b2k + c2 =0        -------(2)

For now the equation reduces to the form

                      dy'/dx' = ( a1x' +b1y' ) / ( a2x' + b2y' )

which is homogeneous in   x'  and  y'  and hence solved .