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MATH 250

Midterm Exam I

Sample 1

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Student Number:

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Section:

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 1

1. (16 points) Answer the questions in the space provided.

a) The following ODE is linear, and k and β are given constants:(dP

dθ− 2θeβθP + θ3

)= kθP

Is this ODE homogeneous?

What is the order of this ODE?

Is P (θ) = 0 a constant solution?

Is this an autonomous ODE?

——————————————————————————————————————

b) In the following ODE, ε is a given constant:

d2Z

da2+ Z = ε1.25a

dZ

da+ ε2 + a

Is this ODE linear or non-linear?

If Z(a) is some solution of this ODE, must 10Z(a) also be a solution?

——————————————————————————————————————

c) Consider the following ODE:df

dx= 3

d2f

dx2+ e8f

How many constant solutions does this ODE have (if any)?

Suppose f1(x) is a solution to this ODE, must 2f1(x) also be a solution?

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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= 6xy2 y(0) = 1

What is the interval of definition of the solution?

Are there any constant solutions?

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3. (12 points) The ODE below is linear, state whether it is homogeneous or non-homogeneous,and find the general solution (assume t > 0):

y′ + 2ty = 2t3

What is the interval of definition of the solution (Hint: It does not depend on the value of c)?

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4. (10 points) The ODE below is linear, state wether it is homogeneous or non-homogeneous, andsolve the ODE with given initial condition.

y′ + 6ty = 36t y(0) = 0

Are there any constant solutions?

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5. (12 points) For the autonomous ODE y′ = y2(y + 1)

a) (4 pts) Find all critical points (if any) and classify their stability.

b) (4 pts) Determine all values of y for which solutions are increasing.

c) (2 pts) What is the limit as t→∞ of a solution y(t) satisfying the initial condition y(−6) = 5?

d) (2 pts) Is a solution y(t) satisfying the initial condition y(−4) = −6 increasing, decreasing,or neither?

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6. (26 points) Show all work.

a) Solve the initial value problem: y′′ − y′ − 6y = 0 y(0) = 1, y′(0) = 2

b) Solve the initial value problem: 4y′′ + 4y′ + y = 0 y(0) = 0, y′(0) = 7

c) Find 2 fundamental solutions to the ODE and prove they form a fundamental set on (−∞,∞)using the Wronskian:

y′′ + 3y′ + 2y = 0

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7. (4 points) y1 = cos 2x and y2 = sin 2x are both solutions to y′′ + 4y = 0. Compute theWronskian of y1 and y2 and clearly state if they form a fundamental set of solutions.

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8. (8 points) The function y1(t) = 79t2 is a solution to the ODE:

t2y′′ + 2ty′ − 6y = 0 t > 0

a) Find a second solution y2 to this ODE which forms a fundamental set with y1.

b) Use the Wronskian to show that your y2 forms a fundamental set with y1.

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