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Differential Equations

Multiple Choice Questions

211.

The solution of the differential equation $(x + 2 y^{3}) \frac{dy}{dx} = Y$ is

y³ + Cx = y
$\frac{{xy}^{4}}{2} + xy = Cy$
y³ + Cy = x
x + 2y³ = y + C

212.

The solution ofthe differential equation

$\frac{dy}{dx} = \frac{y (\log (y) - \log (x) + 1)}{x}$ is

x = ye^cy
y = xe^cy
x = ye^cx
None of these

213.

The solution of the differential equation $\frac{dy}{dx} + \sin (\frac{y + x}{2}) + \sin (\frac{y - x}{2}) = 0$ is

$\log (\tan (\frac{y}{2})) = C - 2 \sin (x)$
$\log (\tan (\frac{y}{4})) = C - 2 \sin (\frac{x}{2})$
$\log (\tan (\frac{y}{2} + \frac{π}{4})) = C - 2 \sin (x)$
$\log (\tan (\frac{y}{2} + \frac{π}{4})) = C - 2 \sin (\frac{x}{2})$

214.

The solution of differential equation $(x^{2} + y^{2}) - 2 xy \frac{dy}{dx} = 0$ is

$x^{2} + y^{2} = xC$
$x^{2} - y^{2} = xC$
x² + y² = C
x² - y² = C

215.

The solution of differential equation $\frac{dy}{dx} = \frac{x (2 \log (x) + 1)}{\sin (y) + ycos (y)}$ is

$ysin (y) = x^{2} \log (x) + C$
$y = x^{2} + \log (x) + C$
$ysin (y) = x^{2} + C$
None of these

216.

Solution of $2 ysin (x) \frac{dy}{dx} = 2 \sin (x) \cos (x) - y^{2} \cos (x), x = \frac{π}{2}$ , y = 1 is given by

y² = sin(x)
y = sin²(x)
y² = cos(x) + 1
None of these

217.

Solution of $x^{2} \frac{dy}{dx} - xy = 1 + \cos (\frac{y}{x})$ is

$\tan (\frac{y}{2 x}) = C - \frac{1}{2 x^{2}}$
$\tan (\frac{y}{x}) = C + \frac{1}{x}$
$\cos (\frac{y}{x}) = 1 + \frac{C}{x}$
$x^{2} = (C + x^{2}) \tan (\frac{y}{x})$

218.
The solution of $\frac{dy}{dx} = \cos (x) (2 - ycsc (x))$ where y = 2, when x = $\frac{π}{2}$ is

$y = \sin (x) + \csc (x)$

$y = \tan (\frac{x}{2}) + \cot (\frac{x}{2})$

$y = \frac{1}{\sqrt{2}} \sec (\frac{x}{2}) + \sqrt{2} \cos (\frac{x}{2})$

None of the above

$y = \sin (x) + \csc (x)$

$Given, \frac{dy}{dx} = \cos (x) (2 - ycsc (x)) \frac{dy}{dx} = 2 \cos (x) - ycot (x) \Rightarrow \frac{dy}{dx} + ycot (x) = 2 \cos (x) IF = e^{\int \cot (x) dx} = \sin (x) ysin (x) = \int 2 \cos (x) . \sin (x) dx + C ysin (x) = \sin^{2} (x) + C at y = 2 and x = \frac{π}{2} \Rightarrow C = 1 ∴ ysin (x) = \sin^{2} (x) + 1 [∵ from Eq . (i)] \Rightarrow y = \sin (x) + \csc (x)$