MATH 250

Midterm Exam I

Sample 2

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A

This exam has 8 questions for a total of 100 points.

Check that your exam has all 8 questions.

In order to obtain full credit, all work must be shown.

You may not use a calculator, cell phone, or computer on this exam.You may not use any notes or books on this exam.

MATH 250 Exam I - Sample 2

1. (16 points) Write the fill word ”True” if the statement is true, or ”False” if it is false in thespace provided, not just ”T” or ”F”.

a) The ODE y′′ − 3ey′+ 4t3y = 0 is 2nd order, linear, homogeneous.

b) The ODE y′ − 9y = 0 is 1st order, linear, and autonomous.

c) The ODE y2y′ = 3 cos t is 1st order, autonomous.

d) The ODE y′′ − (7et cos t)y′ − 3esin ty = 4 is 2nd order, linear.

e) y(t) = 4 is a stable critical point for the ODE y′ = y − 4.

f) y(t) = 0 is a solution to the ODE y′′ + 2ty = et sin t + t3.

g) y(t) = 3et is a solution to the ODE y′ = 3y.

h) y1(t) = et and y2(t) = cos t form a fundamental set of

solutions on (−∞,∞) to the ODE y′′ + y = 0.

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MATH 250 Exam I - Sample 2

2. (12 points) For the ODE below, state whether it is

I linear or non-linear

II autonomous or non-autonomous

III separable or non-separable

and then find an explicit solution to the initial value problem (IVP)

dy

dx=

√1− y2 y(0) = 1

Determine the largest interval of definition of this solution?

What are the constant solutions (if any)?

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MATH 250 Exam I - Sample 2

3. (12 points) Solve the IVP’s below:

a) y′ + 2ty = 3t + 7te−t2

sin t with initial condition y(0) =1

2

b) ty′ − 9y = t2 (assume t > 0) with initial condition y(1) = 1

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MATH 250 Exam I - Sample 2

4. (10 points) For the autonomous ODE y′ = y(y − 5)(2− y)

a) (6 pts) Find all critical points (if any) and classify their stability.

b) (2 pts) Determine all values of y for which solutions are increasing.

c) (2 pts) Is a solution y(t) satisfying the initial condition y(0) = 0 increasing, decreasing, orneither?

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MATH 250 Exam I - Sample 2

5. (12 points) For the autonomous ODE y′ = e−y

a) (2 pts) Find all critical points (if any) and classify their stability.

b) (2 pts) Determine all values of y for which solutions are increasing.

c) (2 pts) Determine all values of y for which solutions are decreasing.

d) (6 pts) Solve the ODE explicitly.

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MATH 250 Exam I - Sample 2

6. (18 points) Show all work.

a) Solve the initial value problem: y′′ + 2y = 0 y(0) = 0, y′(0) = 2

b) Solve the initial value problem: 2y′′ + 3y′ = 0 y(0) = 6, y′(0) = 7

c) Solve the initial value problem: y′′ − 4y′ + 4y = 0 y(0) = 2, y′(0) = 2

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MATH 250 Exam I - Sample 2

7. (10 points) y1 = t is a solution to t2y′′ + 2ty′ − 2y = 0. Assume t > 0 for this problem. Findanother solution y2 which forms a fundamental set with y1. Use the Wronskian to directlyprove the 2 solutions do form a fundamental set on (0,∞).

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MATH 250 Exam I - Sample 2

8. (10 points) The function y1(t) = 6 is a solution to the ODE y′′ − 2

ty′ = 0. Assume t > 0

for this problem. Find another solution y2 which forms a fundamental set with y1. Use theWronskian to directly prove the 2 solutions do form a fundamental set on (0,∞).

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