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