Question

A solution curve of the differential equation (x² + xy + 4x + 2y + 4)dy/dx - y² = 0, x > 0, passes through the point (1, 3). The solution curve 

(A) intersects y = x + 2 exactly at one point;

(B) intersects y = x + 2 exactly at two points;

(C) intersects y = (x + 2)²;

(D) does not intersect y = (x + 3)².

Answer 1

Answer is (A), (D)

See fig.

The given differential equation is

[x² + 4x + 4 + y(x + 2)]dy/dx - y² =  0 (x > 0)

which is further simplified as follows:

[(x + 2)² + y(x + 2)]dy/dx  - y² = 0

Substituting x + 2 = t, we get

dx/dy = dt/dy

which passes through the point (1, 3). Therefore, from Eq. (1), we get

That is, the solution curve intersects y = (x + 2) exactly at one point and not at two points. 

Therefore, option (A) is correct and option (B) is incorrect. 

Checking for option (C), we have

which meets at two points for x < 0 and for x > 0, there is no intersection point (Fig.).

Hence, option (C) is incorrect.

Checking for option (D), we have

Therefore, there is no intersection point for x > 0.