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Differential Equations

Multiple Choice Questions

201.

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

202.

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

203.

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

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

204.

The solution of the differential equation

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

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

205.
The differential equation of all parabolas whose axis is Y-axis, is

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

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

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

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

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

$Axis of parabola = Y axis and vertex of parabola is (0, k) . ∴ Equation of parabola is {(x - 0)}^{2} = 4 a (y - k) x^{2} = 4 ay - 4 ak On differentiate both sides w . r . t,' x', we get 2 x = 4 a \frac{dy}{dx} x = 2 a \frac{dy}{dx} ∴ \frac{1}{2 a} = \frac{1}{x} \frac{dy}{dx} On differentiating both sides w . r . t' x', we get \frac{d}{dx} (\frac{1}{x} . \frac{dy}{dx}) = \frac{d}{dx} (\frac{1}{2 a}) \Rightarrow \frac{1}{x} . \frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} (- \frac{1}{x^{2}}) = 0 ∴ x \frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} = 0$

206.

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

xy = 4
x²y = 4
xy² = 4
x²y² = 4

207.

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

208.

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

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

209.

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

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

210.

The solution of the differential equation $\frac{dy}{dx} = 2 e^{x - y} + x^{2} e^{- y}$ is

$e^{y} = 2 e^{x} + \frac{x^{3}}{3} + C$
$e^{- y} = 2 e^{x} + \frac{x^{- 3}}{3} + C$
$e^{- y} = 2 e^{x} + \frac{x^{3}}{3} + C$
$e^{y} = 2 e^{- x} + \frac{x^{3}}{3} + C$

Previous Year Papers

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Multiple Choice Questions

205. The differential equation of all parabolas whose axis is Y-axis, is xd2ydx2 - dydx = 0 xd2ydx2 + dydx = 0 d2ydx2 - y = 0 d2ydx2 - dydx = 0

205.
The differential equation of all parabolas whose axis is Y-axis, is

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

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

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

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