Solve the following equations:

Given 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 variables on same side we get,

⇒

⇒

⇒

⇒

We know that:

and

Also,

Integrating on both sides, we get,

⇒

⇒

⇒

Since y = vx, we get,

⇒

⇒

⇒

(∵)

⇒

⇒

(∵ alogx = logx^a)

⇒

⇒

∴ The solution for the given Differential equation is