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04:0033–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).
y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ
17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.
b. In what regions are solutions increasing? Decreasing?
y'(t) = (y−1)(1+y)
Properties of stirred tank solutions
b. Verify that M(0) = M₀
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The solution of a stirred tank initial value problem always approaches a constant as t→∞
Blowup in finite time Consider the initial value problem y'(t) = yⁿ + 1, y(0) = y₀, where n is a positive integer.
b. Solve the initial value problem with n = 2 and y₀ = 1/√2.
38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.
b. Sketch the direction field, for t≥0.
y′(t) = 2y + 4
