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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 : 40
    5 x 1 = 5
  1. The differential equation of the family of curves y = Aex + Be−x, where A and B are arbitrary constants is

    (a)

    \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +y=0\)

    (b)

    \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0\)

    (c)

    \(\frac { { d }y }{ { dx } } +y=0\)

    (d)

    \(\frac { { d }y }{ { dx } } -y=0\)

  2. The solution of the differential equation 2x\(\frac{dy}{dx}-y=3\)represents

    (a)

    straight lines

    (b)

    circles

    (c)

    parabola

    (d)

    ellipse

  3. The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is

    (a)

    \(\frac { x }{ { e }^{ \lambda } } \)

    (b)

    \(\frac { { e }^{ \lambda } }{ x } \)

    (c)

    \({ \lambda e }^{ x }\)

    (d)

    ex

  4. The solution of (x2-ay)dx=(ax-y2)dy is

    (a)

    y=x2+y2-a(x+y)

    (b)

    y=x2+y2-a(x+y)

    (c)

    x3+y2=3ayx+c

    (d)

    (x2-ay)(ax-y2)=0

  5. The transformation y=vx reduces \(\\ \\ \\ \frac { dy }{ dx } =\frac { x+y }{ 3x } \)

    (a)

    \(\frac { 3av }{ 4v+1 } =\frac { dx }{ x } \)

    (b)

    \(\frac { 3dv }{ v+1 } =\frac { dx }{ x } \)

    (c)

    \(2x\frac { dv }{ dx } =v\)

    (d)

    \(\frac { 3dv }{ 1-2v } ==\frac { dx }{ x } \)

  6. 5 x 2 = 10
  7. A differential equation, determine its order, degree (if exists)
    \(\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0\)

  8. A differential equation, determine its order, degree (if exists)
    \(y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } } \)

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

  10. Solve: x\(\frac{dy}{dx}=x+y\)

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

  12. 5 x 3 = 15
  13. Determine the order and degree (if exists) of the following differential equations: 
    dy + (xy − cos x)dx = 0

  14. Show that y = 2(x2−1)+Ce−x2 is a solution of the differential equation \(\frac { dy }{ dx } +2xy-4{ x }^{ 3 }=0\)

  15. Solve \((1+{ 2e }^{ x/y })dx+2{ e }^{ x/y }\left( 1-\frac { x }{ y } \right) dy=0\)

  16. Form the differential equation for y=e-2x [A coos 3x-B sin 3x]

  17. Solve: \(\frac{dy}{dx}+y=cos x\)

  18. 2 x 5 = 10
  19. Solve y ' = sin2 (x − y + )1.

  20. Solve:\(\frac { dy }{ dx } \) =(3x+y+4)2.

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