Differential Equations Practice Exams

Time: 1 hour


Problem 1. Find the solution to:

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Problem 2. Solve the initial value problem

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Problem 3. The fox squirrel is a small mammal native to the Rocky Mountains. These squirrels are very territorial:

The carrying capacity N indicates what population is too big, and the sparsity parameter M indicates when the population is too small. A mathematical model which will agree with the above assumptions is the modified logistic model:

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1.
Find the equilibrium (critical) points. Classify them as : source, sink or node. Justify your answers.
2.
Sketch the slope-field.
3.
Assume N=100 and M=1 and k = 1. Sketch the graph of the solution which satisfies the initial condition y(0)=20.
4.
Assume that squirrels are emigrating (from a certain region) with a fixed rate E. Write down the new differential equation.
Discuss the equilibrium (critical) points under the parameter E. When do you observe a bifurcation?

Problem 4. Consider the autonomous differential equation tex2html_wrap_inline50 where the graph of f(y) is


1.
Sketch the Slope Fields of this differential equation
Hint: the graph of the solutions and the graph of f(y) are two different entities!
2.
Sketch the graph of the solution to the IVP

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Find the

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3.
Sketch the graph of the solution to the IVP

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Find the

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If you would like to check your answers, click on Answer.


[CyberExam] [Calculus] [Differential equations]

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