The differential equation of the family of circles through (0,2) and (0,-2) is:

Question

The differential equation of the family of circles through $(0,2)$ and $(0,-2)$ is:

Accepted Answer

Family: $x^2+y^2+\lambda x-4=0$. Differentiating: $2x+2y y'+\lambda=0 \implies \lambda=-2x-2yy'$. Substituting back:
$$x^2+y^2-2x^2-2xyy'-4=0 \implies 2xyy'+x^2-y^2+4=0.$$

Solutions · 1