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Ordinary Differential Equations 2 Mark Book Back Question Paper With Answer Key

12th Standard

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Maths

Time : 00:30:00 Hrs
Total Marks : 84

    2 Marks

    42 x 2 = 84
  1. For each of the following differential equations, determine its order, degree (if exists)
    \(\frac { dy }{ dx } +xy=cotx\)

  2. For each of the following differential equations, determine its order, degree (if exists)
    \({ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ \frac { 2 }{ 3 } }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +4=0\)

  3. For each of the following differential equations, determine its order, degree (if exists)
    \({ { \left( \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) } }^{ 2 }+{ \left( \frac { dy }{ dx } \right) }^{ 2 }=xsin\left( \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) \)

  4. For each of the following differential equations, determine its order, degree (if exists)
    \(\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0\)

  5. For each of the following differential equations, determine its order, degree (if exists)
    \(y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } } \)

  6. For each of the following differential equations, determine its order, degree (if exists)
    \({ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0\)

  7. For each of the following differential equations, determine its order, degree (if exists)
    \({ \left( \frac { d^2y }{ dx^2 } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) } \)

  8. For each of the following differential equations, determine its order, degree (if exists)
    \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } =xy+cos\left( \frac { dy }{ dx } \right) \)

  9. For each of the following differential equations, determine its order, degree (if exists)
    \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +\int { ydx } ={ x }^{ 3 }\)

  10. For each of the following differential equations, determine its order, degree (if exists)
    \(x={ e }^{ xy\left( \frac { dy }{ dx } \right) }\)

  11. Express each of the following physical statements in the form of differential equation.
    (i) Radium decays at a rate proportional to the amount Q present.
    (ii) The population P of a city increases at a rate proportional to the product of population and to the difference between 5,00,000 and the population.
    (iii) 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.
    (iv) A saving amount pays 8% interest per year, compounded continuously. In addition, the income from another investment is credited to the amount continuously at the rate of Rs. 400 per year.

  12. Express each of the following physical statements in the form of 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.

  13. Express each of the following physical statements in the form of differential equation.
    A saving amount pays 8% interest per year, compounded continuously. In addition, the income from another investment is credited to the amount continuously at the rate of Rs. 400 per year.

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

  15. Find the differential equation of the family of all nonhorizontal lines in a plane.

  16. Form the differential equation of all straight lines touching the circle x2 + y2 = r2.

  17. Find the differential equation of the family of circles passing through the origin and having their centres on the x -axis.

  18. Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis.

  19. Find the differential equation of the family of parabolas with vertex at (0, −1) and having axis along the y-axis.

  20. Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin.

  21. Find the differential equation corresponding to the family of curves represented by the equation y = Ae8x + Be-8x, where A and B are arbitrary constants.

  22. Find the differential equation of the curve represented by xy = aex + be−x + x2.

  23. Show that each of the following expressions is a solution of the corresponding given differential equation.
    y = 2x2; xy' = 2y

  24. Show that each of the following expressions is a solution of the corresponding given differential equation.
    y = aex + be−x; y − y = 0

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

  26. Find value of m so that the function y = emx is a solution of the given differential equation, y''− 5y' + 6y = 0

  27. The slope of the tangent to the curve at any point is the reciprocal of four times the ordinate at that point. The curve passes through (2, 5). Find the equation of the curve.

  28. Determine the order and degree (if exists) of the following differential equations: 
    \(\frac { dy }{ dx } =x+y+5\)

  29. Determine the order and degree (if exists) of the following differential equations: 
    \({ \left( \frac { { d }^{ 4 }y }{ { dx }^{ 4 } } \right) }^{ 3 }+4{ \left( \frac { dy }{ dx } \right) }^{ 7 }+6y=5cos3x\)

  30. Determine the order and degree (if exists) of the following differential equations: 
    \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \)

  31. Determine the order and degree (if exists) of the following differential equations: 
    \(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)

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

  33. Find the differential equation for the family of all straight lines passing through the origin.

  34. Form the differential equation by eliminating the arbitrary constants A and B from y = A cos x + B sin x.

  35. Find the differential equation of the family of parabolas y2 = 4ax, where a is an arbitrary constant.

  36. Find the differential equation of the family of all ellipses having foci on the x -axis and centre at the origin.

  37. Show that x+ y2 = r2, where r is a constant, is a solution of the differential equation \(\frac { dy }{ dx } \) = -\(\frac { x }{ y } \).

  38. Show that y = mx + \(\frac{7}{m}\), m ≠ 0 is a solution of the differential equation xy'+7\(\frac{1}{y'}\)-y = 0.

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

  40. Show that y = a cos(log x) + bsin (log x), x > 0 is a solution of the differential equation x2 y" + xy'+y = 0.

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

  42. Solve \({ y }^{ 2 }+{ x }^{ 2 }\frac { dy }{ dx } =xy\frac { dy }{ dx } \)

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