EM-III

1) which of the following is the general solution to d²y /dx² + 3dy /dx − 10y = 0?

In each case, A and B are arbitrary constants. Exactly one option must be correct)

• y = Ae^2x + Be^−5x

• y = Ae^−2x + Be^5x

• y = Ae^2x + Be^5x

• y = Ae−2x + Be−5x

1) Which of the following is the general solution to d²y /dt² - 4dy/ dt +4y = 0?

In each case, A and B are arbitrary constants. Exactly one option must be correct)

• y = Ae^2t + Bte^2t

• y = Ae^2t + Be^−2t

• y = Ae^2t + Be^−6t

• y = Ae(−2+22)t + Be(−2−22)t

1) Which of the following are general solutions to d²x/ dt² − 4dx/ dt + 13x = 0?

In each case, A and B are arbitrary constants.

• x = Ae^5t + Be^−t

• x = Ae^(2+3i)t + Be^(-2−3i)t

• x = e^2x(Acos3x + Bsin3x)

• x = e^3x(Acos2x + Bsin2x)

1) Which of the following is the general solution to d²y/dx² + 10dy/dx + 25y = 0?

In each case, A and B are arbitrary constants. Exactly one option must be correct)

• y = Ae^−5x + Be^−5x

• y = Axe^−5x + Be^−5x

• y = Ae^5x + Be^5x

• y = Axe^5x + Be^5x

1) Consider the differential equation d²y /dx² − 49y = 0.

Which of the following options is correct? Exactly one option must be correct)

• The roots of the auxiliary equation are 0 and 7.

• There is no auxiliary equation for a differential equation of this type.

• The auxiliary equation has a repeated root of 7.

• The roots of the auxiliary equation are 7 and −7

The general solution to d²z /dt² + 6dz/dt + 9z = 0 is

z = Ate^−3t + Be^−3t. Which of the following options is correct?

• As t →∞, z → A for any value of B.

• The behaviour of z as t →∞ depends on the values

  of A and B.

• As t →∞, z → 0 for any values of A and B.

• As t →∞, z →∞ for any values of A and B.

Find the general solution to d²y/dt² − 3dy/dt − 4y = 0 satisfying y(0) = 0 and y′(0) = 5.

• y = e^t − e^−4t

• y = e^−t − e^4t

• y = e^−4t − e^t

• y = e^4t − e^−t

Find a solution to d²x/ dt² + dx/dt − 2x = 0 which satisfies x(0) = 3 and does not tend to infinity (or minus infinity) as t →∞.

• x = 3e^−t

• x = 4e^−2t − e^t

• x = 3e^−2t

• x = 2e^−2t + e^−t

1) If y = Ae3x +Be^2x is a solution to d²y /dx² − 5dy /dx + ky = 0, what is the value of k?

• k = 3

• k = 4

• k = 5

• k = 6