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Ordinary Differential Equations Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  1. The general solution of the differential equation \(\frac { dy }{ dx } =\frac { y }{ x } \) is

    (a)

    xy = k

    (b)

    y = k log x

    (c)

    y = kx

    (d)

    log y = kx

  2. If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\)When

    (a)

    p < q

    (b)

    p = q

    (c)

    p>q

    (d)

    p exists and q does not exist

  3. The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}=3x^2\) and it passes through (-1,1). Then the equation of the curve is

    (a)

    y=x3+2

    (b)

    y=3x2+4

    (c)

    y=3x4+4

    (d)

    y=3x2+5

  4. The I.F. of cosec x\(\frac{dy}{dx}+y\)sec2x=0 is

    (a)

    esec x

    (b)

    etan x

    (c)

    esec x tan x

    (d)

    esec2 x

  5. The differential equation associated with the family of concentric circles having their centres at the origin is _________.

    (a)

    \(\frac { dy }{ dx } =\frac { -x }{ y } \)

    (b)

    \(\frac { dy }{ dx } =\frac { -y }{ x } \)

    (c)

    \(\frac { dy }{ dx } =\frac { x }{ y } \)

    (d)

    \(\frac { dy }{ dx } =\frac { y }{ x } \)

  6. 2 x 2 = 4
  7. If \(\frac { dy }{ dx } =\frac { x-y }{ x+y } \) then 
    1) xdy+y dx=x dx+y dy
    2) \(\int { d(xy) } =\int { xdx } +\int { ydy } \)
    3) x2-y2+2xy=c
    4) x2-y2-2xy=c

  8. Solution of \(\frac{dy}{dx}\)+mx=0 where m<0 is
    1) \(\frac{dy}{dx}\)=-m dy
    2) y=cemx
    3) log x=-my+log x
    4) x=ce-my

  9. 6 x 2 = 12
  10. A differential equation, determine its order, degree (if exists)
    \(\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0\)

  11. A differential equation, determine its order, degree (if exists)
    \({ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0\)

  12. Find value of m so that the function y = emx is a solution of the given differential equation.
    y '+ 2y = 0

  13. Determine the order and degree (if exists) of the following differential equations: 
    dy + (xy − cos x)dx = 0

  14. Solve the differential equation:
    \(\\ \\ \\ \frac { dy }{ dx } =\sqrt { \frac { 1-{ y }^{ 2 } }{ 1-{ x }^{ 2 } } } \)

  15. Solve: \(\frac{dy}{dx}=1+e^{x-y}\)

  16. 3 x 3 = 9
  17. Express the physical statement in the form of the differential equation.
    For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature.

  18. Find the differential equation of the family of all non-vertical lines in a plane.

  19. Solve: x\(\frac{dy}{dx}\)+2y=x2

  20. 4 x 5 = 20
  21. Solve the differential equation:
    (ydx-xdy)cot\(\left( \frac { x }{ y } \right) \)=ny2 dx

  22. Solve the differential equation:
    tan y\(\frac{dy}{dx}\)=cos(x+y)+cos(x-y)

  23. Solve the Linear differential equation:
    \((1+x+{ xy }^{ 2 })\frac { dy }{ dx } +(y+{ y }^{ 3 })=0\)

  24. A population grows at the rate of 2% per year. How long does it take for the population to double?

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