Solve the above equation to find its general solution.

Question: Solve the above equation to find its general solution.

 

The correct answer is –

To solve the above equation, we need to separate the variables v and x. Dividing both sides by (v2 – 1), we get:

dv/dx + (2v/y)/(v2 – 1) = 0

Multiplying both sides by (1/v), we get:

(1/v) dv/dx + (2/y) / (v2 – 1) = 0

Substituting u = 1/v, we get:

du/dx – (2/y)/(1 – u2) = 0

Separating the variables u and x, we get:

∫ (1 – u2) du = ∫ (2/y) dx

Solving the integrals, we get:

u = c1 cos(x/c2) v = 1/u = c2/(c1 cos(x/c2)) y = vx = c2x/(c1 cos(x/c2))

Therefore, the general solution of the given differential equation is y = (c2/c1) x sec(x/c2)