Differential Equations Answers

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

y' − y = 2 cos(4t),  y(0) = 0

y(t) =

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

y' + 3y = e^5t,  y(0) = 2

y(t) =

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?