In each of the question, show that the given differential equation is homogeneous and solve each of them.

Question

In each of the question, show that the given differential equation is homogeneous and solve each of them. (x 2 – y 2 )dx + 2xy dy = 0

Philoid · Accepted Answer

Here, putting x = kx and y = ky

= k⁰.f(x,y)

Therefore, the given differential equation is homogeneous.

To solve it we make the substitution.

y = vx

Differentiating eq. with respect to x, we get

Integrating both sides, we get

Put 1 + v² = t

2vdv = dt

log(t)

∴ log(1 + v²) = -logx + logC (∴ From (i) eq.)

The required solution of the differential equation.