Find u substitute in y’-(y/x)=y^2

Question

I am having trouble identifying where to start to find the U subsitute. 

Answer 1

1st Order Bernoulli :  y' - P(x)* y = Q(X)*Y^N

                               y' - (1/x)*y = 1* y^2

 

1st order Bernoulli with p(x) = 1/x , q(x) = 1, AND n=2

 

Divides by y^2

 

  y'/y^2 - (1/x)(1/y) = 1

 

 y'/y^2 - 1/(xy) = 1

 

Let U = 1/y.  Then dU = -  dy/y^2 --> -dU = dy/y^2

 

-dU - (1/x)U = 1    <---- substitution

 

du +  (1/x)U = -1

 

 

This is a first order linear ODE with P(x) = 1/x and Q(x) = -1

 

Integrating factor :   exp ( integral (1/x)) = exp (  ln (x)) = x

 

x*dU + (1/x)(x) U = -x

 

x * dU + U = -x

 

d/dx [ x* U ] = -x

 

Integraing...

 

 x*U = (-1/2)X^2 + c

 

  U = (1/2)X + c/x = [ (1/2)x^ + c ]/x

 

1/y =  [ (1/2)x^2 + c ]/x

 

y = x/ [ (1/2)x^2 + c]

 

-----------------------------------------

 

Check by differentiation using the quotient rule:

 

 y' = {[(-1/2)x^2 + c] - x [- x ] }/ [ (1/2)x^2 + c]^2

 

    = [(-1/2)x^2 +c + x^2] / [(-1/2)x^2 + c]^2

 

   = [(1/2)x^2 + c]/[(-1/2)x^2 + c]^2

 

-------------------------------------------------------

 

y' - (1/x)y =

 

[(1/2)x^2 + c]/[(-1/2)x^2 + c]^2 - 1 / [(-1/2)x^2 + c] =

 

 

 

  {[(1/2)x^2 + c] - [(-1/2)x^2 + c] }/[(-1/2)x^2 + c]^2

 =     X^2 / [ (-1/2)X^2 + C]^2  

 

 = Y^2