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Differential Equations

Multiple Choice Questions

561.

The solution of differential equation

$4 xy \frac{dy}{dx} = \frac{3 {(1 + x)}^{2} (1 + y^{2})}{(1 + x^{2})}$ is

$\log (1 + y) = \log (x) + 2 \tan (x) + C$
$\log (1 + y^{2}) = 3 \log (\frac{1}{x}) + 6 \tan^{- 1} (x) + C$
$\log (1 + y^{2}) = 3 \log (x) + 6 \tan^{- 1} (x) + C$
None of the above

562.

The order and power of differential equation

$\frac{d^{2} y}{{dx}^{2}} + 7 \frac{dy}{dx} + \int ydx = \sin (x)$ is

1, 3
3, 1
1, 2
2, 1

563.

The solution of differential equation ${xcos}^{2} (y) dx = {ycos}^{2} (x) dx$ is

$xtan (x) - ytan (y) - \log (\sec (x) / \sec (y)) = c$
$ytan (x) - xtan (x) - \log (\sec (x) . \sec (y)) = c$
$xtan (x) - ytan (y) + \log (\sec (x) . \sec (y)) = c$
None of the above

564.

Differential equation of those circles which passes through origin and their centres lie on y-axis will be

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

565.

The differential equation of all circles touching the axis of y at origin and centre on the x-axis is given by

$xy \frac{dy}{dx} - x^{2} + y^{2} = 0$
$2 xy \frac{dy}{dx} - x^{2} - y^{2} = 0$
$(x^{2} + y^{2}) \frac{dy}{dx} - 2 xy = 0$
None of these

566.

The solution of the differential equation

$(e^{- 2 \sqrt{x}} - \frac{y}{\sqrt{x}}) \frac{dx}{dy} = 1$ is given by

$e^{2 \sqrt{x}} = 2 \sqrt{x} + c$
${ye}^{- 2 \sqrt{x}} = \sqrt{x} + c$
$y = \sqrt{x}$
None of these

567.
The solution of the differential equation
$\frac{dy}{dx} = \sqrt{\frac{1 - y^{2}}{1 - x^{2}}}$ is

$\sin^{- 1} (y) - \sin^{- 1} (x) = c$

$\sin^{- 1} (y) + \sin^{- 1} (x) = c$

$\sin^{- 1} (xy) = 2$

None of these

$\sin^{- 1} (y) - \sin^{- 1} (x) = c$

$Given, \frac{dy}{dx} = \sqrt{\frac{1 - y^{2}}{1 - x^{2}}} \frac{dy}{\sqrt{1 - y^{2}}} = \frac{dx}{\sqrt{1 - x^{2}}} On integrating both sides, we get \sin^{- 1} (y) - \sin^{- 1} (x) = c$

568.

The solution of differential equation (1 + x)ydx + (1 - y)x dy = 0 is

$\log_{e} (xy) + x - y = c$
$\log_{e} (\frac{x}{y}) + x + y = c$
$\log_{e} (\frac{x}{y}) - x + y = c$
$\log_{e} (xy) - x + y = c$

569.

The differential equation of all circles which passes through the origin and whose centre lies on y-axis is

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

570.

The general solution of y²dx + (x² - xy + y²)dy = 0 is

$\tan^{- 1} (\frac{x}{y}) + \log (y) + c = 0$
$2 \tan^{- 1} (\frac{x}{y}) + \log (x) + c = 0$
$\log (y + \sqrt{x^{2} + y^{2}}) + \log (y) + c = 0$
$\sinh^{- 1} (\frac{x}{y}) + \log (y) + c = 0$

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Multiple Choice Questions

567. The solution of the differential equationdydx = 1 - y21 - x2 is sin-1y - sin-1x = c sin-1y + sin-1x = c sin-1xy = 2 None of these

567.
The solution of the differential equation
$\frac{dy}{dx} = \sqrt{\frac{1 - y^{2}}{1 - x^{2}}}$ is

$\sin^{- 1} (y) - \sin^{- 1} (x) = c$

$\sin^{- 1} (y) + \sin^{- 1} (x) = c$

$\sin^{- 1} (xy) = 2$

None of these