Engineering Mathematics - Differential Equations

Question - 1

The solution of the first order differential equation \(x^{ \prime }(t)=-3x(t),x(0)={ x }_{ 0 }\) is

  • A \(x(t)={ x }_{ 0 }e^{ -3t }\)
  • B \(x(t)={ x }_{ 0 }e^{ -3 }\)
  • C \({ x }_{ 0 }e^{ -1/3 }\)
  • D \({ x }_{ 0 }e^{ -t }\)

Question - 2

For the equation x"(t)+3x'+2x(t) = 5, the solution x(t) approaches which of the following values as \(t\rightarrow \infty \)?

  • A 0
  • B 5/2
  • C 5
  • D 10

Question - 3

The order and degree of the differential equation \(\frac { { d }^{ 3 }y }{ dx^{ 2 } } +4\sqrt { \left( \frac { dy }{ dx } \right) ^{ 3 }+{ y }^{ 2 }=0 } \) are respectively

  • A 3 and 2
  • B 2 and 3
  • C 3 and 3
  • D 3 and 1

Question - 4

The general solution of \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +y=0\) is

  • A y = P cos x + Q sin x
  • B y = P cos x
  • C y = P sin x
  • D y = P \(sin^{ 2 }x\)

Question - 5

The degree of the differential equation \(\frac { { d }^{ 2 }x }{ d{ t }^{ 2 } } +2{ x }^{ 3 }=0\) is

  • A 0
  • B 1
  • C 2
  • D 3

Question - 6

The solution for the differential equation \(\frac { dy }{ dx } ={ x }^{ 2 }y\) with the condition that y = 1 at x = 0 is

  • A \(y={ e }^{ 1/2x }\)
  • B \(logy=\frac { { x }^{ 3 } }{ 3 } +4\)
  • C \(logy=\frac { { x }^{ 2 } }{ 2 } \)
  • D \(y=e^{ x^{ 3 }/3 }\)

Question - 7

The solution of the differential equation \(\frac { dy }{ dx } +\frac { y }{ x } =x\) with the condition that y = 1 at x = 1 is

  • A \(y=\frac { 2 }{ { 3x }^{ 2 } } +\frac { x }{ 3 } \)
  • B \(y=\frac { x }{ 2 } +\frac { 1 }{ 2x } \)
  • C \(y=\frac { 2 }{ 3 } +\frac { x }{ 3 } \)
  • D \(y=\frac { 2 }{ 3x } +\frac { { x }^{ 2 } }{ 3 } \)

Question - 8

The solution of the ordinary differential equation \(\frac { d^{ 2 }y }{ dx^{ 2 } } +\frac { dy }{ dx } -6y=0\) is

  • A \(y=c_{ 1 }e^{ 3x }+c_{ 2 }e^{ -2x }\)
  • B \(y=c_{ 1 }e^{ 3x }+c_{ 2 }e^{ 2x }\)
  • C \(c_{ 1 }e^{ -3x }+c_{ 2 }e^{ 2x }\)
  • D \(c_{ 1 }e^{ -3x }+c_{ 2 }e^{ -2x }\)

Question - 9

Th solution of differential equation \(\frac { dy }{ dx } =-\frac { x }{ y } \) at x=1 and \(y=\sqrt { 3 } \) is

  • A \(x-{ y }^{ 2 }=-2\)
  • B \(x+{ y }^{ 2 }=-4\)
  • C \({ x }^{ 2 }-{ y }^{ 2 }=2\)
  • D \({ x }^{ 2 }+{ y }^{ 2 }=4\)

Question - 10

The solution of the differential equation \({ x }^{ 2 }\frac { dy }{ dx } +2xy-x+1=0\) given that at x=1, y=0 is

  • A \(\frac { 1 }{ 2 } -\frac { 1 }{ x } +\frac { 1 }{ { 2x }^{ 2 } } \)
  • B \(\frac { 1 }{ 2 } -\frac { 1 }{ x } -\frac { 1 }{ { 2x }^{ 2 } } \)
  • C \(\frac { 1 }{ 2 } +\frac { 1 }{ x } +\frac { 1 }{ { 2x }^{ 2 } } \)
  • D \(-\frac { 1 }{ 2 } +\frac { 1 }{ x } +\frac { 1 }{ { 2x }^{ 2 } } \)
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