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:
⇒
⇒ -log(1-v2) = logx + logC
⇒ log(1-v2)-1 = log(Cx)
(∵ alogx = logxa)
(∵ loga + logb = logab)
⇒
Applying exponential on both sides, we get,
⇒
Since y = vx, we get,
⇒
⇒
⇒
⇒
Cross multiplying on both sides we get,
⇒ x = C(x2 – y2)
∴ The solution for the given Differential equation is x = C(X2-y2)