Show that the differential equation is homogeneous and solve it.

Given, We have a differential equation,

To Find: Prove that it is a homogenous differential equation and solve.

Explanation: We have

It can be written as

Now, to prove that it is a Homogenous differential equation, we put x = λx and y = λy then

Now, Taking λ as common from both numerator and denominator, we get

If, then it is an homogenous differential equation

Now, Solution of this differential equation is

Put y = v x

Then,

So, When we compare this derivative from the given equation we get

Taking x as common from R.H.S

On integrating both sides,

For Solving L.H.S

Put v² + v + 1 = t

2v + 1 dv = dt

So,

Now, Put the value of v = y/ x, we get

Hence, This is the solution of Given differential equation.