A solution curve of the differential equation (x² + xy + 4x + 2y + 4)dy/dx - y² = 0, x > 0, passes through the point (1, 3). Then the solution curve - Does NOT intersect y = (x + 3)²; Intersects y = x + 2 exactly at one point; Intersects y = (x + 2)²

[(x+2)(x+2+y)]dy/dx - y² = 0. Substitute y = (x+2)t ⇒ dy/dx = (x+2)dt/dx + t ⇒ (x+2)² = 0 or (1+t)((x+2)dt/dx + t) - t² = 0 ⇒ (x+2)(1+t)dt/dx + t = 0 (1+t)/t dt = -dx/(x+2) ln t + t = -ln(x+2) + c ⇒ ln(y/(x+2)) + (y/(x+2)) = -ln(x+2) + c ln y - ln(x+2) + y/(x+2) = -ln(x+2) + c ⇒ ln y + y/(x+2) = c ln 3 + 3/3 = +c ⇒ c = ln 3 + 1 ⇒ ln y + y/(x+2) = ln 3e (A) ln y + y/(x+2) = ln 3e ⇒ ln(x+2) + 1 = ln 3 + 1 ⇒ one solution (D) y = (x+3)² ⇒ ln(x+3)² + (x+3)²/(x+2) = ln 3 + 1 2ln(x+3) + (x+2)² + 1 + 2(x+2)/(x+2) = ln 3 + 1 g(x) = 2ln(x+3) + (x+2) + 2 + 1/(x+2) - ln 3 ≉ g'(x) = 2/(x+3) + 1 - 1/(x+2)² = 2/(x+2)² - (x+3)(x+3)/(x+2)² + 1 > 0 Since x > 0 given, and g(0) > 0, therefore g(x) will never intersect X-axis when x > 0. Hence option (D).