The differential equation 2xy , , dy = (x^2 + y^2 + 1) dx determi… - Vidyadip

MCQ

The differential equation $2xy\,\, dy = (x^2 + y^2 + 1) dx$ determines

A
A family of circles with centre on $x-$ axis
B
A family of circles with centre on $y-$ axis
✓
A family of rectangular hyperbola with centre on $x-$ axis
D
A family of rectangular hyperbola with centre on $y-$ axis

✓

Answer

Correct option: C.

A family of rectangular hyperbola with centre on $x-$ axis

c
$2 x y d y=\left(x^{2}+1\right) d x+y^{2} d x$

$\Rightarrow \frac{2 x y d y-y^{2} d x}{x^{2}}=\left(1+\frac{1}{x^{2}}\right) d x$

$\Rightarrow d\left(\frac{y^{2}}{x}\right)=d\left(x-\frac{1}{x}\right)$

$\Rightarrow \frac{y^{2}}{x}=x-\frac{1}{x}+c \Rightarrow y^{2}=x^{2}-1+c x$

$\Rightarrow\left(x+\frac{c}{2}\right)^{2}-y^{2}=1+\frac{c^{2}}{4}$

which represents a family of rectangular hyperbolas with centre on $\mathrm{x}$ -axis.

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