Differential Equations Important Questions

12th Standard EM

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Business Maths

Time : 01:00:00 Hrs
Total Marks : 50
    10 x 1 = 10
  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 integrating factor of the differential equation \(\frac{dx}{dy}+Px=Q\)

    (a)

    eഽPdx

    (b)

    eഽPdx

    (c)

    ഽPdy

    (d)

    eഽPdy

  3. 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

  4. 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

  5. Solution of \(\frac { dy }{ dx } \) + Px = 0

    (a)

    x=cepy

    (b)

    x=ce−py

    (c)

    x = py + c

    (d)

    x = cy

  6. The differential equation satisfied by all the straight lines in xy plane is

    (a)

    \(\frac { dy }{ dx } \)=a constant

    (b)

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

    (c)

    y+ \(\frac { dy }{ dx } \) = 0

    (d)

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

  7. If y = k.eλx then its differential equation where k is arbitrary constant is

    (a)

    \(\frac { dy }{ dx } \)=λy

    (b)

    \(\frac { dy }{ dx } \)=ky

    (c)

    \(\frac { dy }{ dx } \)+ky=0

    (d)

    \(\frac { dy }{ dx } \)=eλx

  8. The differential equation obtained by eliminating a and b from y = a e3x + b e-3x is

    (a)

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

    (b)

    \(\frac { { d }^{ 2 }y }{ dx^{ 2 } } \)-9y=0

    (c)

    \(\frac { { d }^{ 2 }y }{ dx^{ 2 } } -9\frac { dy }{ dx } \)

    (d)

    \(\frac { { d }^{ 2 }y }{ dx^{ 2 } } \)+9x=0

  9. The differential equation formed by eliminating A and B from y = ex (A cos x + B sin x) is

    (a)

    y2+y1=0

    (b)

    y2-y1=0

    (c)

    y2-2y1+2y=0

    (d)

    y2-2y1-2y=0

  10. The degree of the differential equation \(\sqrt { 1+\left( \frac { { d }y }{ dx } \right) ^{ \frac { 1 }{ 3 } } } =\frac { { d }^{ 2 }y }{ dx^{ 2 } } \) is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    6

  11. 5 x 1 = 5
  12. y = mx+ c

  13. (1)

    1

  14. y= mx

  15. (2)

    \(e^{ \int { pdy } }\)

  16. Degree of linear differential equation

  17. (3)

    \(\frac { dy }{ dx } \)+Py=Q

  18. General form of linear equation

  19. (4)

    Family of parabolas having origin as vertex

  20. I.F. of \(\frac { dy }{ dx } \)+Px=Q

  21. (5)

    Family of lines

    5 x 2 = 10
  22. Find the differential equation of the family of all straight lines passing through the origin.

  23. Form the differential equation that represents all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x axis.

  24. Find the differential equation of all circles passing through the origin and having their centers on the y axis.

  25. Solve: ydx − xdy = 0

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

  27. 5 x 3 = 15
  28. Find the differential equation of the family of curves \(y=\frac { a }{ x } +b\) where a and b are arbitrary constants

  29. Solve the differential equation \(\frac { dy }{ dx } =\frac { x-y }{ x+y } \)

  30. Solve: (3D2 + D - 14)y = 4 - 13\({ e }^{\frac{-7}{3}x}\)

  31. Suppose that the quantity demanded \({ Q }_{ d }=29-2p-5\frac { dp }{ dt } +\frac { { d }^{ 2 }p }{ { dt }^{ 2 } } \) and quantity supplied Qs= 5 + 4p where p is the price. Find the equilibrium price for market clearance.

  32. Find the differential equation of all circles x2 +y2 + 2gx = 0 which pass through the origin and whose centres are on the X-axis.

  33. 2 x 5 = 10
  34. The marginal cost function of manufacturing x gloves is 6 +10x−6x2. The total  cost of producing a pair of gloves is Rs. 100. Find the total and average cost function.

  35. The net profit p and quantity x satisfy the differential equation \(\frac { dp }{ dx } =\frac { 2{ p }^{ 3 }-{ x }^{ 3 } }{ 3x{ p }^{ 2 } } \). Find the relationship between the net profit and demand given that p = 20, when x = 10.

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