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05:0933–36. {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)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. Euler’s method is used to compute exact values of the solution of an initial value problem.
{Use of Tech} Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M(t) be the mass of a tumor at time t≥0. The relevant initial value problem is
dM/dt=−rM ln(M/K),M(0)=M0,
where r and K are positive constants and 0<M0<K.
b. Solve the initial value problem and graph the solution for r=1,K=4, and M0=1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor?
29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.
b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.
y′(t) = 2t + 1, y(0) = 0; y(t) = t² + t
brOrthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection. A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. Use the following steps to find the orthogonal trajectories of the family of ellipses 2x² + y² = a²
b. The family of trajectories orthogonal to 2x² + y² = a² satisfies the differential equation dy/dx = y/(2x). Why?
