Question

If the general solution of the differential equation y' = y/x + ϕ(x/y) for some function ϕ, is given by y  ln|cx| = x, where c is an arbitrary constant, then ϕ(2) is equal to

(A) 4

(B) 1/4

(C) - 4

(D) - 1/4

Answer 1

Answer is

General solution of dy/dx = y/x + ϕ(x/y) is

yln|cx| = x    (1)

On differentiating y In |cx| = x, we get,

y x 1/|cx| . c + ln |cx|dy/dx = 1

⇒ y/x + x/y(dy/dx) = 1

Therefore,

x/y(dy/dx) = 1 - y/x ⇒ dy/dx  = y/x - y²/x²    (2)

Now comparing with the given differential equation

ϕ(x/y) = y²/x² = - (y/x)² = - (1/(x/y))²