Solve the Differential Equation

Question

dy/dx = (y/x) - tan(y/x)?

Answer 1

The differential equation dy/dx = (y/x) - tan(y/x)

Let y/x = t

y = xt

Differentiating with respect to x.

dy/dx = x dt/dx + t

The equation is dy/dx = (y/x) - tan(y/x)

x dt/dx + t = t - tan(t)

x dt/dx = - tan(t)

- 1/tan(t) dt = (1/x) dx

- cot(t) dt = (1/x) dx

Apply integral on each side.

-ʃcot(t) dt =  ʃ(1/x) dx

- ln(sint) = ln(x) + C

C = - sin(t)/x

Sin(t) = - Cx

Solution sin(y/x) = - Cx.