Differential Equations Answers

Use the Laplace transform to solve the given initial-value problem.

dy/dt  − y = 1,    y(0) = 0

y(t) =

Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your answer as a function of t.)

ℒ⁻¹{(s/(s² + 3s − 4)}

Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your answer as a function of t.)

ℒ⁻¹{(1/s²)  − (720/s⁷)}

Use Theorem 7.1.1 to find ℒ{f(t)}.

 (Write your answer as a function of s.)

f(t) = (t + 1)³

Use Definition 7.1.1.

DEFINITION 7.1.1    Laplace Transform

Let f be a function defined for t ≥ 0.

 Then the integral

ℒ{f(t)} = ∞

e^−stf(t) dt0

is said to be the Laplace transform of f, provided that the integral converges.

Find ℒ{f(t)}.

 (Write your answer as a function of s.)

f(t) = {t, 0 ≤ t < 1

{1,  t ≥ 1

Find ℒ{f(t)}.

 (Write your answer as a function of s.)

f(t) = −1,    0 ≤ t < 11,t ≥ 1

The speed of a boat in still water is 15 km/hr. It needs four more hours to travel 63 km against the current of a river than it needs to travel down the river. Determine the speed of the current of the river.

An object is thrown vertically upward from a height of h0 ft with an initial speed of v0 ft/sec. Its height h (in feet) after t seconds is given by

h= -16t² + h0. Given this, if it is thrown vertically upward from the ground with an initial speed of 64 ft/sec,

(a) At what time will the height of the ball be 15 ft? (two answers)

(b) How long will it take for the ball to reach 63 ft?

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.)

3y'' − 8y' − 3y = 4

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

7x − sin(x) + 20 cos(5x)