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,
⇒
⇒
(∵ log C is an arbitrary constant)
⇒
(∵ alogx = logxa)
⇒
(∵ loga + logb = logab)
Since y = vx,
we get,
⇒
⇒
⇒
Applying exponential on both sides we get,
⇒
⇒
∴ The solution of the Differential equation is