Differential Equations Answers

1. Find linearly independent functions that are annihilated by the given differential operator. (Give as many functions as possible. Use x as the independent variable. Enter your answers as a comma-separated list.)

D4

2. Solve the given differential equation by undetermined coefficients.
y'' + y' + y = x sin x

y(x) = ______

1. Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. (Use D for the differential operator.)

y''' + 14y'' + 49y' = ex

_______ y= e^x

2. Find a linear differential operator that annihilates the given function. (Use D for the differential operator.)

1 + 3x − 4x3

1. Find a linear differential operator that annihilates the given function. (Use D for the differential operator.)

e−x + 5xex − x2ex

2. Find a linear differential operator that annihilates the given function. (Use D for the differential operator.)

cos 6x

1. Solve the given boundary-value problem.
y'' − 16y' + 64y = 0, y(0) = 1, y(1) = 0

y(x) = _____

1. Find a homogeneous linear differential equation with constant coefficients whose general solution is given.

y = c1 cos 2x + c2 sin 2x

A. y'' + 2y = 0
B. y'' − 2y = 0
C. y'' + 4y = 0
D. y'' − 4y = 0
E. y'' − 4y' + 4y = 0

1. Find a homogeneous linear differential equation with constant coefficients whose general solution is given.

y = c1e−x cos x + c2e−x sin x

A. y'' + 2y' + 2y = 0
B. y'' − 2y' + 2y = 0
C. y''' + 2y'' + 2y' = 0
D. y''' − 2y'' + 2y' = 0
E. y'' + 1 = 0

2. Find the general solution of the given second-order differential equation.
y'' + 36y = 0

y(x) = ____

1. Find the general solution of the given higher-order differential equation.
16 d^4 y/ dx^4 + 24 d^2 y/ dx^2+ 9y = 0

y(x) =_____

2. Find the general solution of the given second-order differential equation.
y'' − 10y' + 26y = 0

y(x) = ______

1. Find the general solution of the given second-order differential equation.
3y'' + y' = 0

y(x) = _____

2. Solve the given initial-value problem.
y''' + 14y'' + 49y' = 0, y(0) = 0, y'(0) = 1, y''(0) = −8

y(x) =____

1. Find the general solution of the given higher-order differential equation.
d^3u/ dt^3 + d^2u/ dt^2 − 2u = 0

u(t) = ____

2. Find the general solution of the given higher-order differential equation.
y''' − 9y'' + 15y' + 25y = 0

y(x) =____

In one model of the changing population P(t) of a community, it is assumed that
dP/ dt = dB/ dt - dD/ dt, where dB/dt and dD/dt are the birth and death rates, respectively.

1. Solve for P(t) if dB/dt = k1P and dD/dt = k2P. (Assume P(0) = P0.)
P(t) =___

2. Analyze the cases k1 > k2, k1 = k2, and k1 < k2.
For k1 > k2,one has the following.

A. P → − ∞ as t → ∞
B. P → 0 as t → ∞
C. P → ∞ as t → ∞
D. P = 0 for every t
E. P = P0 for every t

3. For k1 = k2, one has the following.

A. P → − ∞ as t → ∞
B. P → 0 as t → ∞
C. P → ∞ as t → ∞
D. P = 0 for every t
E. P = P0 for every t

4. For k1 < k2, one has the following.

A. P → − ∞ as t → ∞
B. P → 0 as t → ∞
C. P → ∞ as t → ∞
D. P = 0 for every t
E. P = P0 for every t