Exercises In Exercises 1–4: (a) Determine the vector field corresponding to the

Question

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

Explanation / Answer

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)

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