Using y=ux to solve 1st order diff equation

Question

Prove y=ux reduces y'=f(y/x) into separable equations in u and x hence solve

Answer 1

 if  y =  ux,    then y' = x u' + u   and the diff equation is

 

  x u' + u =  f(u)

   x u' =  f(u) -u

 

   dx / x =   du /( f(u) -u  ) so

 

   ln (x)   = ∫ du/( f(u) -u)  + C

 

   x =   Λ exp [ ∫ du/( f(u) -u) ]

       This approach has reduced the problem to quadrature.

     If an anti derivative,   F (u) , exists for   1/( f(u) -u ) 

  x =  Λ exp [ F(u) ]  and

 

 x = Λ exp [  F( y/x) ]

 

This is an implicit equation for y =  y(x)

 

Λ is the constant of integration that must be determined from initial conditions.