Give Differential equation is:

⇒

⇒ ……(1)

Homogeneous equation: A equation is said to be homogeneous if f(zx,zy) = zⁿf(x,y) (where n is the order of the homogeneous equation).

Let us assume:

⇒

⇒

⇒

⇒ f(zx,zy) = z⁰f(x,y)

So, given differential equation is a homogeneous differential equation.

We need a substitution to solve this type of linear equation and the substitution is y = vx.

Let us substitute this in (1)

⇒

We know that

⇒

⇒

⇒

Bringing like coefficients on same sides we get,

⇒

We know that ∫adx = ax + C and

Also,

Integrating on both sides, we get,

⇒

⇒ v = logx + C

Since y = vx,

we get,

⇒

Cross multiplying on both sides we get,

⇒ y = xlogx + Cx

∴ The solution for the given differential equation is y = xlogx + Cx