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Differential Equations

Multiple Choice Questions

541.
The differential equation of the family of circles touching Y-axis at the origin is

$(x^{2} + y^{2}) \frac{dy}{dx} - 2 xy = 0$

$(x^{2} - y^{2}) + 2 xy \frac{dy}{dx} = 0$

$(x^{2} - y^{2}) \frac{dy}{dx} - 2 xy = 0$

$(x^{2} + y^{2}) \frac{dy}{dx} + 2 xy = 0$

$(x^{2} - y^{2}) + 2 xy \frac{dy}{dx} = 0$

Let centre of circle on X-axis be (h, 0),

The radius of circle will be h.

$∴$ The equation of circle having centre (h, 0) and radius h is

${(x - h)}^{2} + {(y - 0)}^{2} = h^{2} \Rightarrow x^{2} + h^{2} - 2 hx + y^{2} = h^{2} \Rightarrow x^{2} - 2 hx + y^{2} = 0 . . . (i) On differentiating both sides w . r . tx, we get 2 x - 2 h + 2 y \frac{dy}{dx} = 0 \Rightarrow h = x + y \frac{dy}{dx} On putting h = x + y \frac{dy}{dx} in Eq . (i), we get x^{2} - 2 (x + y \frac{dy}{dx}) x + y^{2} = 0 \Rightarrow - x^{2} + y^{2} - 2 xy \frac{dy}{dx} = 0 \Rightarrow (x^{2} - y^{2}) + 2 xy \frac{dy}{dx} = 0$

542.

The degree and order of the differential equation ${[1 + {(\frac{dy}{dx})}^{3}]}^{\frac{7}{3}} = 7 (\frac{d^{2} y}{{dx}^{2}})$ respectively are

3 and 7
3 and 2
7 and 3
2 and 3

543.

The particular solution of the differential equation

$y (1 + \log (x)) \frac{d x}{d y} - xlog (x) = 0$ , when, x = e, y = e² is

y = exlog(x)
ey = xlog(x)
xy = elog(x)
ylog(x) = ex

544.

If $\sin (x)$ is the integrating factor (IF) of the linear differential equation $\frac{dy}{dx} + Py = Q$ , then P is

$\log (\sin (x))$
$\cos (x)$
$\tan (x)$
$\cot (x)$

545.

The solution of the differential equation

$\frac{dy}{dx} = \tan (\frac{y}{x}) + \frac{y}{x}$ is

$\cos (\frac{y}{x}) = cx$
$\sin (\frac{y}{x}) = cx$
$\cos (\frac{y}{x}) = cy$
$\sin (\frac{y}{x}) = cy$

546.

The differential equation of all parabolas whose axis is Y-axis, is

$x \frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} = 0$
$x \frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} = 0$
$\frac{d^{2} y}{{dx}^{2}} - y = 0$
$\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} = 0$

547.

The particular solution of the differential equation xdy + 2ydx = 0, when x = 2, y = 1 is

xy = 4
x²y = 4
xy² = 4
x²y² = 4

548.

The general solution of the equation $\frac{dy}{dx} = \frac{y^{2} - x}{2 y (x + 1)}$ is

y² = (1 + x)log(1 + x) - c
$y^{2} = (1 + x) \log (\frac{c}{(1 - x)}) - 1$
$y^{2} = (1 - x) \log (\frac{c}{(1 - x)}) - 1$
$y^{2} = (1 + x) \log (\frac{c}{(1 + x)}) - 1$

549.

The general solution of the differential equation $\frac{dy}{dx} + \sin (\frac{x + y}{2}) = \sin (\frac{x - y}{2})$ is

$\log_{e} |\tan (\frac{y}{2})| = - 2 \sin (\frac{x}{2}) + C$
$\log_{e} |\tan (\frac{y}{4})| = 2 \sin (\frac{x}{2}) + C$
$\log_{e} (\tan (\frac{y}{2})) = - 2 \sin (\frac{x}{2}) + C$
$\log_{e} (\tan (\frac{y}{4})) = - 2 \sin (\frac{x}{2}) + C$

550.

The function y specified implicitly by the relation $\int_{0}^{y} e^{t} dt + \int_{0}^{x} \cos (t) dt$ = 0 satisfies the differential equation

$e^{2 y} (\frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2}) = \sin (x)$
$e^{y} (\frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2}) = \sin (2 x)$
$e^{y} (2 \frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2}) = \sin (x)$
$e^{y} (\frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2}) = \sin (x)$

Previous Year Papers

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Test Yourself

Multiple Choice Questions

541. The differential equation of the family of circles touching Y-axis at the origin is x2 + y2dydx - 2xy = 0 x2 - y2 + 2xydydx = 0 x2 - y2dydx - 2xy = 0 x2 + y2dydx + 2xy = 0

541.
The differential equation of the family of circles touching Y-axis at the origin is

$(x^{2} + y^{2}) \frac{dy}{dx} - 2 xy = 0$

$(x^{2} - y^{2}) + 2 xy \frac{dy}{dx} = 0$

$(x^{2} - y^{2}) \frac{dy}{dx} - 2 xy = 0$

$(x^{2} + y^{2}) \frac{dy}{dx} + 2 xy = 0$