The differential equation of the system of all circles of radius

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 Multiple Choice QuestionsMultiple Choice Questions

91.

The solution of the differential equation dydx = sinx + ytanx +y - 1 is

  • cscx +y + tanx + y = x +c

  • x + cscx + y = c

  • x + tanx + y = c

  • x + secx + y = c


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


93.

The differential equation dydx + y + 3 = 0 admits

  • infinite number of solutions

  • no solutions

  • a unique solution

  • many solutions


94.

Solution of the differential equation xdy - ydx - x2 + y2dx = 0

  • y - x2 + y2 = cx2

  • y + x2 + y2 = cx2

  • y + x2 + y2 = cy2

  • x - x2 + y2 = cy2


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95.

If xsinxydy = ysinyx - xdx and y1 = π2, then the value of cosyx is equal to :

  • x

  • 1x

  • logx

  • ex


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96.

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

  • 1 + dydx32 = r2d2ydx22

  • 1 + dydx32 = r2d2ydx23

  • 1 + dydx23 = r2d2ydx22

  • 1 + dydx23 = r2d2ydx23


C.

1 + dydx23 = r2d2ydx22

The equation of the family of circles of radius r is

(x - a)2 + (y - b)2 = r2       ...(i)

where a and b are arbitrary constants.

On differentiating Eq. (i) w.r.t. x, we get

  2(x - a) + 2(y - b)dydx = 0

 x - a + y - bdydx = 0      ...(ii)

On differentiating Eq. (ii) w.r.t. x, we get

1 + y - bd2ydx2 + dydx2 = 0 y - b = - 1 + dydx2d2ydx2        ...(iii)

On putting the value of (y - b) in Eq. (ii), we get

x - a = 1 + dydx2dydxd2ydx2        ...(iv)

On putting the value of (y - b) and (x - a), in Eq. (i), we get

1 + dydx22dydx2d2ydx22 + 1 + dydx22d2ydx22 = r2 1 + dydx23 = r2d2ydx22This is the required differential equation.


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97.

The general solution of the differential equation

d2ydx2 + 2dydx + y = 2e3x is given by

  • y = c1 + c2xex + e3x8

  • y = c1 + c2xe- x + e- 3x8

  • y = c1 + c2xe- x + e3x8

  • y = c1 + c2xex + e- 3x8


98.

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

  • xy = 13y3 + c

  • xy = y4 + c

  • y4 = 4xy + c

  • 4y = y3 + c


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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

  •  s3

  •  1s3

  •  s5


100.

The solution of the differential equation dydx = yx + ϕyxϕ'yx is

  • yx = k

  • ϕyx = kx

  • yx = k

  • ϕyx = ky


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