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Differential Equations

Multiple Choice Questions

91.

The solution of the differential equation $\frac{d y}{d x} = \sin (x + y) \tan (x + y) - 1$ is

$\csc (x + y) + \tan (x + y) = x + c$
$x + \csc (x + y) = c$
$x + \tan (x + y) = c$
$x + \sec (x + y) = c$

92.
The differential equation of all straight lines touching the circle x² + y² = a² is

${(y - \frac{dy}{dx})}^{2} = a^{2} [1 + {(\frac{dy}{dx})}^{2}]$

${(y - x \frac{dy}{dx})}^{2} = a^{2} [1 + {(\frac{dy}{dx})}^{2}]$

$(y - x \frac{dy}{dx}) = a^{2} [1 + \frac{dy}{dx}]$

$(y - \frac{dy}{dx}) = a^{2} [1 - \frac{dy}{dx}]$

${(y - x \frac{dy}{dx})}^{2} = a^{2} [1 + {(\frac{dy}{dx})}^{2}]$

The equation of straight line touching the circle x² + y² = a is

$xcos (θ) + ysin (θ) = a$ ...(i)

On differentiating w.r.t. x, regarding $θ$ as a constant

$\cos (θ) + y' \sin (θ) = 0$ ...(ii)

From Eqs. (i) and (ii), we get

$\cos (θ) = \frac{ay'}{xy' - y} and \sin (θ) = - \frac{a}{xy' - y} ∵ \cos^{2} (θ) + \sin^{2} (θ) = 1 ∴ \frac{a^{2} y'^{2} + a^{2}}{{(xy' - y)}^{2}} = 1 \Rightarrow {(y - x \frac{dy}{dx})}^{2} = a^{2} [1 + {(\frac{dy}{dx})}^{2}]$

93.

The differential equation $|\frac{dy}{dx}| + |y| + 3 = 0$ admits

infinite number of solutions
no solutions
a unique solution
many solutions

94.

Solution of the differential equation $xdy - ydx - \sqrt{x^{2} + y^{2}} dx = 0$

$y - \sqrt{x^{2} + y^{2}} = {cx}^{2}$
$y + \sqrt{x^{2} + y^{2}} = {cx}^{2}$
$y + \sqrt{x^{2} + y^{2}} = {cy}^{2}$
$x - \sqrt{x^{2} + y^{2}} = {cy}^{2}$

95.

If $xsin (\frac{x}{y}) dy = [ysin (\frac{y}{x}) - x] dx$ and $y (1) = \frac{π}{2}$ , then the value of $\cos (\frac{y}{x})$ is equal to :

x
$\frac{1}{x}$
$\log (x)$
e^x

96.

The differential equation of the system of all circles of radius r in the xy plane is :

${[1 + {(\frac{dy}{dx})}^{3}]}^{2} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{2}$
${[1 + {(\frac{dy}{dx})}^{3}]}^{2} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{3}$
${[1 + {(\frac{dy}{dx})}^{2}]}^{3} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{2}$
${[1 + {(\frac{dy}{dx})}^{2}]}^{3} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{3}$

97.

The general solution of the differential equation

$\frac{d^{2} y}{{dx}^{2}} + 2 \frac{d y}{d x} + y = 2 e^{3 x}$ is given by

$y = (c_{1} + c_{2} x) e^{x} + \frac{e^{3 x}}{8}$
$y = (c_{1} + c_{2} x) e^{- x} + \frac{e^{- 3 x}}{8}$
$y = (c_{1} + c_{2} x) e^{- x} + \frac{e^{3 x}}{8}$
$y = (c_{1} + c_{2} x) e^{x} + \frac{e^{- 3 x}}{8}$

98.

The solution of the differential ydx + (x - y³)dy = 0 is:

xy = $\frac{1}{3} y^{3} + c$
xy = y⁴ + c
y⁴ = 4xy + c
4y = y³ + c

99.

If the distances covered by a particle in time t is proportional to the cube root of its velocity, then the acceleration is

a constant
$\propto s^{3}$
$\propto \frac{1}{s^{3}}$
$\propto s^{5}$

100.

The solution of the differential equation $\frac{dy}{dx} = \frac{y}{x} + \frac{ϕ (\frac{y}{x})}{ϕ' (\frac{y}{x})}$ is

$xϕ (\frac{y}{x}) = k$
$ϕ (\frac{y}{x}) = kx$
$yϕ (\frac{y}{x}) = k$
$ϕ (\frac{y}{x}) = ky$

Previous Year Papers

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Multiple Choice Questions

92. The differential equation of all straight lines touching the circle x2 + y2 = a2 is y - dydx2 = a21 + dydx2 y - xdydx2 = a21 + dydx2 y - xdydx = a21 + dydx y - dydx = a21 - dydx