Problems

In Problems 1-4, compute the Euler approximation to the given initial value problem with Δt = 0.1 for 0 ≤ t ≤ 1.

5           Show that the initial value problem y' = 1 + ty², y(0) = 0 has a unique solution v(t) for 0 ≤ t ≤ j.

6           Show that the initial value problem ½(0) = 0 has a unique solution y(t) for 0 ≤ t ≤ ½.

7           Show that the initial value problem y' = arctan (t + e^y), v(0) = 4 has a unique solution y(t) for 0 ≤ t < ∞.

8           Show that the initial value problem y' = t + exp(-y²), y(0) = 1 has a unique solution y(t) for 0 ≤ t < ∞.

9           Show that the initial value problem y' = y^1/3, y(0) = 0 has infinitely many solutions for 0 ≤ t < ∞.

10            Show that the initial value problem y' = t(|1 - y²|)^1/2, jy(0) = 1 has infinitely many solutions for 0 ≤ t < ∞.

11            Suppose that f(t, y) is continuous for all t and y. Prove that for each point (a, y₀), the initial value problem y' = cos (f(t, y)), y(a) = y₀ has a solution y(t), a ≤ t < ∞.

12            Suppose that f(t, y) and g(t) are continuous for all t and y and that |f(t, y)| ≤ g(t) for all t and y. Prove that for each point (a, y₀) the initial value problem y' = f(t, y), y(a) = y₀ has a solution y(t) for o ≤ t < ∞.

13            Suppose that f(t, y) is continuous for all t and y. Prove that for each point (a, y₀) there is a number b > a such that the initial value problem y' = f(t,y), y(a) = y₀ has a solution y(t), a ≤ t ≤ b.