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Differential Equations

Multiple Choice Questions

391.

The degree of the differential equation $x = 1 + (\frac{d y}{d x}) + \frac{1}{2!} {(\frac{d y}{d x})}^{2} + \frac{1}{3!} {(\frac{d y}{d x})}^{3} + . . .$

3
2
1
not defined

392.

Solution of the differential equation xdy - ydx = 0 represents a

parabola
circle
hyperbola
straight line

393.

The general solution of the differential equation

$\frac{d y}{d x} = e^{y + x} + e^{y - x}$ is

e^{- y} = e^x - e^{- x} + c
e^{- y} = e^{- x} - e^x + c
e^{- y} = e^x + e^-^x + c
e^y = e^x + e^{- x} + c

394.

The order of the differential equation

$\frac{d^{2} y}{d x^{2}} = \sqrt{1 + {(\frac{dy}{dx})}^{2}}$ is

395.

The degree of the differential equation

${[1 + {(\frac{dy}{dx})}^{2}]}^{\frac{5}{3}} = \frac{d^{2} y}{d x^{2}}$ is

1
5
$\frac{10}{3}$
3

396.

The differential equation of all parabolas whose axes are parallel to y-axis, is

$\frac{d^{3} y}{d x^{3}} = 0$
$\frac{d^{2} y}{d x^{2}} = 0$
$\frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} = 0$
$\frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} + y = 0$

397.

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

e^{- y} = e^x - e^{- x} + c, c integrating constant
e^{- y} = e^- ^x - e^x + c, c integrating constant
e^{- y} = e^x + e^{- x} + c, c integrating constant
e^{- y} + e^x - e^{- x} = c, c integrating constant

398.
If $x = e^{t} \sin (t), y = e^{t} \cos (t)$ , then $\frac{d^{2} y}{d x^{2}}$ at $x = π$ is

$2 e^{π}$

$\frac{1}{2} e^{π}$

$\frac{1}{2 e^{π}}$

$\frac{2}{e^{π}}$

$\frac{2}{e^{π}}$

$Since, x = e^{t} \sin (t) and y = e^{t} \cos (t) On differentiating w . r . t . t respectively, we get \frac{d x}{d t} = e^{t} \cos (t) + \sin (t) e^{t} and \frac{d y}{d t} = - e^{t} \cos (t) + \sin (t) e^{t} ∴ \frac{d y}{d x} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{e^{t} (\cos (t) - \sin (t))}{e^{t} (\cos (t) + \sin (t))} = \frac{\cos (t) - \sin (t)}{\cos (t) + \sin (t)}$

$Again differentiating, we get \frac{d^{2} y}{d x^{2}} = \frac{[(\cos (t) + \sin (t)) (- \sin (t) - \cos (t)) - (\cos (t) - \sin (t)) (- \sin (t) + \cos (t))]}{{(\cos (t) + \sin (t))}^{2}} \frac{dt}{dx} = \frac{- {(\sin (t) + \cos (t))}^{2} - {(\cos (t) - \sin (t))}^{2}}{{(\cos (t) + \sin (t))}^{2}} \times \frac{1}{e^{t} (\cos (t) + \sin (t))} = - [\frac{\sin^{2} (t) + \cos^{2} (t) + 2 \sin (t) \cos (t) + \cos^{2} (t) + \sin^{2} (t) - 2 \sin (t) \cos (t)}{e^{t} {(\cos (t) + \sin (t))}^{3}}] = - \frac{2}{e^{t} {(\cos (t) + \sin (t))}^{3}} \Rightarrow {(\frac{d^{2} y}{d x^{2}})}_{(x = π)} = \frac{- 2}{e^{π} {(\cos (π) + \sin (π))}^{3}} = \frac{2}{e^{π}}$