Spring 2002 Math 308 Exam: Differential Equations, Exams of Differential Equations

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Math 308 – Differential Equations Exam 1 Spring 2002

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# Points Score

Total 100

Reminder: (^) ∫ dx x^2 + 1

= arctan(x) + C

1. Consider the differential equation

dy dt

= f (y),

where f (y) is shown in the following plot:

f(y)

–1.

–0.

0

–3 –2 –1 1 2 3 y

(The graph of f is tangent to the y axis at y = −2.)

(a) Sketch the phase line for this equation. (You may draw it in the above plot.)

(b) Identify the equilibria, and determine whether each equilibrium is a source, a sink, or a node.

2. For each of the following differential equations, classify it as linear or not, autonomous or not, and separable or not. Then find the general solution.

(a)

dy dt

= 2y + 1

Linear: YES / NO Autonomous: YES / NO Separable: YES / NO

(b)

dy dt

= 2y + e^2 t

Linear: YES / NO Autonomous: YES / NO Separable: YES / NO

(c)

dy dt

= ty^2 + 4t − y^2 − 4

Linear: YES / NO Autonomous: YES / NO Separable: YES / NO

4. Find the general solution to dy dt

t

y + t^4 ,

and show that for all solutions, lim t→ 0 y(t) = 0. Explain why this does not contradict the Uniqueness Theorem.

5. (a) Suppose the initial value problem

dy dt

= f (y), y(0) = 1, has the solution y(t) = √ 2 t + 1. What is the solution if the initial condition is changed to y(0) = 2? Briefly explain how you found your answer.

(b) The plot below shows a direction field for an autonomous system of two first order differential equations dx/dt = f (x, y), dy/dt = g(x, y). The solid curve corresponds to a solution (x(t), y(t)) to this system, with initial conditions x(0) = 1 and y(0) = 2. In the same plot, sketch the curve given by (x(t − 1), y(t − 1)).

–0.

0

1

2

y

–0.5 0.5 1 1.5 2 x

(b) Use a qualitative analysis to determine the behavior of the solution to the initial value problem in (a). Include a rough sketch of v(t) vs. t, and describe what happens to v(t) as t → ∞. Does the tank overflow?

7. Consider the following differential equation

dy dt

= (μ − y)y,

where μ is a parameter. Describe how the number and type of equilibria depend on the parameter μ. Sketch the bifurcation diagram for this system, and determine the bifurcation value. Sketch three phase lines, one for μ less than, equal to, and greater than the bifurcation value.

Phase Planes 1 2

0

1

2

3

y

–3 –2 –1 1 2 3 x

0

1

2

3

y

–3 –2 –1 1 2 3 x

0

1

2

3

y

–3 –2 –1 1 2 3 x

0

1

2

3

y

–3 –2 –1 1 2 3 x

0

1

2

3

y

–3 –2 –1^1 2 x

Graphs of x(t) and y(t) versus t A B

0

1

2

3

–2 –1 1 2 t

0

1

2

3

–2 –1 1 2 t

C D

0

1

2

3

–2 –1 1 2 t

0

1

2

3

–2 –1 1 2 t

E

0

1

2

3

–2 –1 1 2 t