Question

1. By hand, sketch the slope field for the DE x ′ = x(1 − x/4)...

1. By hand, sketch the slope field for the DE x ′ = x(1 − x/4) in the window 0 ≤ t ≤ 8, 0 ≤ x ≤ 8 at the integer lattice points. What is the value of the slope field along the lines x = 0 and x = 4? Show that x(t) = 0 and x(t) = 4 are constant solutions to the DE.

2. Draw several isoclines of the differential equation x ′ = x 2 + t 2 , and from your plots determine, approximately, the graphs of the solution curves.

3. Draw the nullclines for the equation x ′ = 1 − x 2 . Graph the isoclines, or the locus of points in the plane where the slope field is equal to −3 and +3.

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Answer #1

2)

Given that  

the differential equation ==>x ′ = x 2 + t 2

A few isoclines are given below:

From this, the graphs of the solution curves are approximately:

Since for some constant

3)

Nullclines of which are given below

Isoclines where the slope field is equal to -3 and +3 is given below:

which is impossible and

The region where the slope field is between -3 and +3 is given below:

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