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Differential Equations Model Question Paper

12th Standard

    Reg.No. :
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Business Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  1. The differential equation \({ \left( \frac { dx }{ dy } \right) }^{ 3 }+2{ y }^{ \frac { 1 }{ 2 } }\) = x is ______.

    (a)

    of order 2 and degree 1

    (b)

    of order 1 and degree 3

    (c)

    of order 1 and degree 6

    (d)

    of order 1 and degree 2

  2. The complementary function of (D2+ 4)y = e2x is ______.

    (a)

    (Ax +B)e2x

    (b)

    (Ax +B)e−2x

    (c)

    A cos 2x + B sin 2x

    (d)

    Ae−2x+ Be2x

  3. The particular integral of the differential equation is \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -8\frac { dy }{ dx } \) + 16y = 2e4x ______.

    (a)

    \(\frac { { x }^{ 2 }{ e }^{ 4x } }{ 2! } \)

    (b)

    \(\frac { { e }^{ 4x } }{ 2! } \)

    (c)

    x2e4x

    (d)

    xe4x

  4. The integrating factor of x \(\frac { dy }{ dx } \) - y = x2 is ______.

    (a)

    \(\frac {-1}{x}\)

    (b)

    \(\frac {1}{x}\)

    (c)

    log x

    (d)

    x

  5. The solution of the differential equation \(\frac { dy }{ dx } \) + Py = Q where P and Q are the function of x is ______.

    (a)

    \(y=\int Q e^{\int P d x} d x+c\)

    (b)

    \(y=\int Q e^{-\int P d x} d x+c\)

    (c)

    \(y e^{\int P d x}=\int Q e^{\int P d x} d x+c\)

    (d)

    \(y e^{\int P d x}=\int Q e^{-\int P d x} d x+C\)

  6. 5 x 2 = 10
  7. Find the differential equation of all circles passing through the origin and having their centers on the y axis.

  8. Find the differential equation of the family of parabola with foci at the origin and axis along the x-axis.

  9. Solve: cosx(1 + cos y)dx − sin y(1 + sin x)dy = 0

  10. Solve: (1 − x)dy − (1 + y)dx = 0

  11. Solve the following homogeneous differential equations.
    \(x\frac { dy }{ dx } =x+y\).

  12. 5 x 3 = 15
  13. Find the differential equation of the family of straight lines y = mx + c when
    (i) m is the arbitrary constant
    (ii) c is the arbitrary constant
    (iii) m and c both are arbitrary constants.

  14. Find the particular solution of the differential equation x2 dy + y(x + y)dx = 0 given  that x = 1, y = 1

  15. If the marginal cost of producing x shoes is given by (3xy + y2)dx + (x+ xy)dy = 0 and the total cost of producing a pair of shoes is given by Rs. 12. Then find the total cost  function.

  16. Solve \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -4\frac { dy }{ dx } +5y\) = 0

  17. Solve \(\frac { { d }^{ 2 }x }{ d{ t }^{ 2 } } -\frac { 3dx }{ dt } +2x\) = 0 given that when t = 0, x = 0 and \(\frac { dx }{ dt } \) = 1

  18. 4 x 5 = 20
  19. Solve \(\frac { dy }{ dx } +\frac { y }{ x } ={ x }^{ 3 }\)

  20. Solve cos2\(\frac{dy}{dx}\) + y = tan x

  21. Equipment maintenance and operating costs (are related to the overhaul interval x by the equation \({ x }^{ 2 }\frac { dc }{ dx } -10xc=-10\) with c = c0 and x = x0. Find c as a function of x.

  22. Suppose that the quantity needed Qd = 42 -4p-4\(\frac { dp }{ dt } +\frac { { d }^{ 2 }p }{ { dt }^{ 2 } } \) and quantity supplied Q= -6 + 8p where p is the price. Find the s equilibrium price for market clearance.

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