Use y=ux to solve

Question

Use the substitution y=ux to solve

dy/dx=y/x–(x/y)e^(y²/x²)

Answer 1

If y = ux, then the differential equation can be rewritten as:

 

   u + x(du/dx) = u - (1/u)e^{u^2}

 

So, x(du/dx) = -(1/u)e^{u^2}

 

Separating variables, we have -ue^{-u^2}du = (1/x)dx

 

-∫ue^{-u^2}du = ∫(1/x)dx

 

Let w = -u², then dw = -2udu.  So, -udu = (1/2)dw.

 

Thus, (1/2)∫e^wdw = lnlxl+C₁

 

                e^w = ln(x²) + C  (C = 2C₁)

 

Since w = -u², e^{-u^2} = ln(x²) + C

 

But, u = y/x gives us e^{-(y^2/x^2)} = ln(x²) + C