Complex Roots in Differential Equations Quiz

What is the characteristic polynomial for a second-order linear homogeneous differential equation?

• aM^2 + bM + c

• aM^3 + bM^2 + cM

• aM + b

• aM^2 + bM^2 + c

In the general solution form for complex roots, what does the alpha represent?

• The coefficient of the cosine function

• The imaginary part of the solution

• The real part of the solution

• The coefficient of the sine function

How is the beta used in the general solution form for complex roots?

• As a constant in the characteristic polynomial

• As a coefficient of the y-prime term

• As a constant multiple in the sine and cosine functions

• As a constant multiple in the exponential function

What is the solution for M when the characteristic polynomial is 1M^2 - 2M + 5 = 0?

• 1 ± 2i

• 2 ± 3i

• 0 ± 2i

• 1 ± 3i

In the example y'' + 4y = 0, what are the values of alpha and beta?

• Alpha = 0, Beta = 3

• Alpha = 2, Beta = 0

• Alpha = 0, Beta = 2

• Alpha = 1, Beta = 2

For the equation 4M^2 + 3 = 0, what is the value of M?

• ±i/2

• ±i√3

• ±2i

• ±i√3/2

In the example 2y'' - 2y' + 5y = 0, what is the simplified form of the solution?

• e^(x)(c1 cos(3x/2) + c2 sin(3x/2))

• e^(x/2)(c1 cos(2x) + c2 sin(2x))

• e^(x)(c1 cos(2x) + c2 sin(2x))

• e^(x/2)(c1 cos(3x/2) + c2 sin(3x/2))

What does Euler's formula state?

• e^(iθ) = cos(θ) + i sin(θ)

• e^(iθ) = sin(θ) + i cos(θ)

• e^(iθ) = sin(θ) - i cos(θ)

• e^(iθ) = cos(θ) - i sin(θ)

How are linear combinations used in deriving the general solution form?

• By subtracting the sine terms

• By multiplying the cosine terms

• By combining real and imaginary parts to form real-valued functions

• By adding the exponential terms

What is the final form of the general solution for complex roots?

• y = e^(βx)(c1 cos(αx) + c2 sin(αx))

• y = e^(βx)(c1 sin(αx) + c2 cos(αx))

• y = e^(αx)(c1 cos(βx) + c2 sin(βx))

• y = e^(αx)(c1 sin(βx) + c2 cos(βx))