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The spring 2002 exam for math 308: differential equations. The exam covers topics such as phase lines, equilibria, separable differential equations, and initial value problems. Students are required to sketch phase lines, identify equilibria, find general solutions, and solve initial value problems.
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Math 308 โ Differential Equations Exam 1 Spring 2002
Name:
Reminder: (^) โซ dx x^2 + 1
= arctan(x) + C
dy dt
= f (y),
where f (y) is shown in the following plot:
f(y)
โ1.
โ0.
0
โ3 โ2 โ1 1 2 3 y
(The graph of f is tangent to the y axis at y = โ2.)
(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.
(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 โ y^2 โ 4
Linear: YES / NO Autonomous: YES / NO Separable: YES / NO
t
y + t^4 ,
and show that for all solutions, lim tโ 0 y(t) = 0. Explain why this does not contradict the Uniqueness Theorem.
dy dt
= f (y), y(0) = 1, has the solution y(t) = โ 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 โ 1), y(t โ 1)).
โ0.
0
1
2
y
โ0.5 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 โ โ. Does the tank overflow?
dy dt
= (ฮผ โ y)y,
where ฮผ is a parameter. Describe how the number and type of equilibria depend on the parameter ฮผ. Sketch the bifurcation diagram for this system, and determine the bifurcation value. Sketch three phase lines, one for ฮผ less than, equal to, and greater than the bifurcation value.
Phase Planes 1 2
0
1
2
3
y
โ3 โ2 โ1 1 2 3 x
0
1
2
3
y
โ3 โ2 โ1 1 2 3 x
0
1
2
3
y
โ3 โ2 โ1 1 2 3 x
0
1
2
3
y
โ3 โ2 โ1 1 2 3 x
0
1
2
3
y
โ3 โ2 โ1^1 2 x
Graphs of x(t) and y(t) versus t A B
0
1
2
3
โ2 โ1 1 2 t
0
1
2
3
โ2 โ1 1 2 t
0
1
2
3
โ2 โ1 1 2 t
0
1
2
3
โ2 โ1 1 2 t
0
1
2
3
โ2 โ1 1 2 t