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Differential Equations

Multiple Choice Questions

221.

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

222.

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

223.

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

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

224.

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

225.
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

None of these

$Given differential equation is, (e^{- 2 \sqrt{x}} - \frac{y}{\sqrt{x}}) \frac{dx}{dy} = 1 This differential equation can be rewritten as \frac{dy}{dx} + \frac{y}{\sqrt{x}} = e^{- 2 \sqrt{x}} Here, P = \frac{1}{\sqrt{x}}, Q = e^{- 2 \sqrt{x}} ∴ IF = e^{\int \frac{1}{\sqrt{x}} dx} = e^{2 \sqrt{x}} ∴ Solution is = \int e^{2 \sqrt{x}} e^{- 2 \sqrt{x}} dx = \int 1 dx \Rightarrow y e^{2 \sqrt{x}} = x + c$