Question

The curve satisfying the differential equation, ydx – (x + 3y2)dy = 0 and passing through the point (1, 1) also passes through the point

(A) (1/4, 1/2)

(B) (1/4, -1/2)

(C) (1/3, - 1/3)

(D) (-1/3, 1/3)

Answer 1

Answer is (D) (-1/3, 1/3)

The given differential equation is

The above differential equation is of the form

dx/dy + P(y)x = Q(y)

with P(y) = -1/y and Q(y) = 3y

To solve the differential equation of this form, let us find the integrating factor:

Solution of integral has the form

For point (1, 1): 1 = 3 + C ⇒ C = −2

Therefore, x = 3y2 – 2y.

This equation is satisfied by point (-1/3, 1/3)

-1/3 = 3 x (1/3)2 = 2(1/3)

⇒ -1/3 = 1/3 - 2/3

= -1/3