Question

A curve passing through $(2,3)$ and satisfying the differential equation $\int\limits_0^x t y(t) d t=x^2 y(x),(x>0)$ is -

• A. $x^2+y^2=13$
• B. $y^2=\frac{9}{2} x$
• C. $\frac{x^2}{8}+\frac{y^2}{18}=1$
• D. $x y=6$

Answer 1

Answer: (D)

$\int\limits_0^x t y(t) d t=x^2 y(x)$
Differentiating, we get
$x y=2 x y+x^2 \frac{d y}{d x} \Rightarrow x^2 \frac{d y}{d x}+x y=0$
$x \frac{d y}{d x}+y=0 \Rightarrow x d y+y d x=0$
$d(x y)=0 \Rightarrow x y=c$
$\therefore \text { since it passes through }(2,3) $
$\therefore \text { c }=6$
$\text { Hence } x y=6 .$