Show that the differential equation representing one parameter family of curves (x² – y²) = c(x² + y²)² is (x³ – 3xy²)dx = (y³ – 3x²y)dy.

Given, A differential equation (x³ – 3xy²)dx = (y³ – 3x²y)dy

To Prove: Prove that (x² – y²) = c(x² + y²)²

Explanation: We have (x³ – 3xy²)dx = (y³ – 3x²y)dy

Now, Find dy/dx from the given equation

Let

Put x = λx and y = λy in F(x, y)

Taking λ³ as a common from numerator and denominator then,

F(x, y) = λ⁰F(x, y)

Since, F(x, y) is a homogenous function of degree zero.

Let us Assume , y = vx - - - (ii)

Differentiate equation (i) w.r.t x

- - - (iii)

Now, Comparing the equation (i) and (iii), we get

Taking x³ as common then, we get

Now, Integrating both sides

Let I = , then

I = log|x| + C

I =

Let 1 - v⁴ = t

Differentiating , - 4v³ dv = dt

And,

Now, put u = v²

On differentiating w.r.t v

We know,

Putting the values of t

Now, Putting v = y/x

Hence, Proved