# 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

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