Consider the initial value problem if 0 < t < 1 { if 1 < t < 7 if 7 < t < ∞, y' + 5y = (formulas) 0 9 a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). y(t) 0 b. Solve your equation for Y(s). Y(s) = L{y(t)} = y(0) = 3. = = c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). help

-2 df Find the solution of the differential equation dz = z3 + sin(z) with initial condition f (0) = 2. Verify the result by differentiation.

Newton's Law of Cooling tells us that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings. This can be modeled by the differential equation = k(T – A), where T is the temperature of the object after t units of time dt dT have passed, A is the ambient temperature of the object's surroundings, and k is a constant of proportionality. Suppose that a cup of coffee begins at 188 degrees and, after sitting in room temperature of 68 degrees for 18 minutes, the coffee reaches 178 degrees. How long will it take before the coffee reaches 166 degrees? Include at least 2 decimal places in your answer. minutes

Using the Laplace transform, solve the differential equation y" + 4y/ – 5y = el4t such that yo = Y% = 0.