Exercises In Exercises 14 a Determine the vector field corre

Exercises In Exercises 1–4:

(a) Determine the vector field corresponding to the given system.

(b) Sketch the vector field at enough points to get a sense of its geometric structure.

(c) Sketch several typical solutions and briefly describe their behavior.

1. x\' = 0, y\' = 1

2. x\' = 1, y\' = y

3. x\' = x, y\' = y

4. x\' = x -1, y\' = -y + 1

6. Consider the system x\' = 2x

y\' = -y.

(a) Sketch the vector field.

(b) Show that for every solution (x(t), y(t)) there exists a constant C such that the solution lies on the curve xy^2 = C. Find C in terms of x0 = x(0) and y0 = y(0).

7. Find a system of first-order dierential equations in x and y such that the functions x(t) = e^t cost, y(t) = e^t sin t

Solution

Given y(t) = e^t sin t\\

differentiate both sides:

y\'(t) = e^t cos(t) + e^t sin(t)

Put the value of x(t) = e^t cost, y(t) = e^t sin t

hence y\'(t) = x(t) + y(t)

ir y\'(t) - y (t)= x(t)