Question

The equation of the curve through the point $(3,2)$ and whose slope is $\frac{x^2}{y+1}$, is

• A. $\frac{y^2}{2}+y=\frac{x^3}{3}+5$
• B. $y+y^2=x^3-21$
• C. $y^2+2y=\frac{2x^3}{3}-10$
• D. $\frac{y^2}{2}+y=\frac{x^3}{3}-5$

Answer 1

The correct option is D $\frac{y^2}{2}+y=\frac{x^3}{3}-5$

$\frac{dy}{dx}=\frac{x^3}{y+1}$
$(y+1)dy=x^{2}dx$

$\frac{y^2}{2}+y=\frac{x^3}{3}+c$

Passes through $(3,2)$

$\frac{4}{2}+2=\frac{27}{3}+c$
$4=9+c$
$c=-5$
$\frac{y^2}{2}+y=\frac{x^3}{3}-5$