The solution of the differential equation \( \displaystyle\frac{dy}{dx}=\frac{x+2}{3-y},(y\ne 3),y(0)=0 \) represents
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Circle
The variables separate:
\( \displaystyle (3-y)\,dy=(x+2)\,dx \)
Integrating both sides,
\( \displaystyle 3y-\frac{y^2}{2}=\frac{x^2}{2}+2x+C \)
Putting x = 0, y = 0 gives C = 0. Multiplying by 2:
\( \displaystyle 6y-y^2=x^2+4x \)
\( \displaystyle x^2+y^2+4x-6y=0 \)
Completing the squares:
\( \displaystyle (x+2)^2+(y-3)^2=13 \)
This is a circle with centre (−2, 3) and radius \( \sqrt{13} \).