1
MHT CET 2023 9th May Evening Shift
+2
-0

The differential equation of all circles which pass through the origin and whose centres lie on $$\mathrm{Y}$$-axis is

A
$$\left(x^2-y^2\right) \frac{\mathrm{d} y}{\mathrm{~d} x}-2 x y=0$$
B
$$\left(x^2-y^2\right) \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x y=0$$
C
$$\left(x^2-y^2\right) \frac{\mathrm{d} y}{\mathrm{~d} x}+x y=0$$
D
$$\left(x^2-y^2\right) \frac{\mathrm{d} y}{\mathrm{~d} x}-x y=0$$
2
MHT CET 2023 9th May Morning Shift
+2
-0

The differential equation of all parabolas, whose axes are parallel to $$\mathrm{Y}$$-axis, is

A
$$y_3=1$$
B
$$y_3=0$$
C
$$y_3=-1$$
D
$$y y_3+y_1=0$$
3
MHT CET 2023 9th May Morning Shift
+2
-0

The particular solution of the differential equation $$\left(1+y^2\right) \mathrm{d} x-x y \mathrm{~d} y=0$$ at $$x=1, y=0$$, represents

A
circle
B
pair of straight lines
C
hyperbola
D
ellipse
4
MHT CET 2022 11th August Evening Shift
+2
-0

The general solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3 x+y}{x-y}$$ is (where $$C$$ is a constant of integration.)

A
$$\tan ^{-1}\left(\frac{y}{x}\right)+\log \left(\frac{y^2+3 x^2}{x^2}\right)=\log (x)+C$$
B
$$\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{y}{x \sqrt{3}}\right)+\log \left(\frac{y^2+3 x^2}{x^2}\right)^{\frac{1}{2}}=\log (x)+C$$
C
$$\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{y}{x \sqrt{3}}\right)-\log \left(\frac{y^2+3 x^2}{x^2}\right)^{\frac{1}{2}}=\log (x)+C$$
D
$$\tan ^{-1}\left(\frac{x}{y}\right)+\log \left(\frac{y^2+3 x^2}{x^2}\right)=\log (x)+C$$
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