Solve the following equations:

Question

Philoid · Accepted Answer

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,

⇒

(∵ log C is an arbitrary constant)

⇒

(∵ alogx = logx^a)

⇒

(∵ loga + logb = logab)

Since y = vx,

we get,

⇒

Applying exponential on both sides we get,

⇒

∴ The solution of the Differential equation is