Math 2410 Practice Exam I Feb xth, 2019

Name:

Instructions:

• Answer each question to the best of your ability. Show your work or receive no credit.

• All answers must be written clearly. Be sure to erase or cross out any work that you do not wantgraded. Partial credit can not be awarded unless there is legible work to assess.

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Academic Integrity AgreementI certify that all work given in this examination is my own and that, to my knowledge, has not been usedby anyone besides myself to their personal advantage. Further, I assert that this examination was taken inaccordance with the academic integrity policies of the University of Connecticut.

Signed:(full name)

Questions: 1 2 3 4 5 TotalScore:

1

Exam One Key

ExamOneKey

1. (5 points) Solve the linear initial value problem

((x2 � 1) dydx + 2y = (x+ 1)2

y(2) = 0

Hint: Partial fractions may be helpful.

2

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2. (5 points) Consider the ODE

(y3 + kxy4 � 2x)dx+ (3xy2 + 20x2y3)dy = 0,

where k is to be determined.

(a) Find k so that the given ODE is an exact equation.

(b) Using this value of k, solve the given ODE.

3

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3. (5 points) Consider the ODE(xk + yey/x)dx� xey/xdy = 0,

where k is to be determined.

(a) Find k so that the given ODE is a homogeneous equation.

(b) Using this value of k, solve the given ODE.

4

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4. (5 points) Consider the Bernoulli equation

y0 � y = yp,

where 0 < p < 1 is arbitrary.

(a) Solve the given equation with p left arbitrary. The solution will depend on p.

(b) Let yp denote your answer in part (a). Compute

limp!0

yp

and compare the limit to the solution to y0 � y = 1.

(c) (If you have too much time on your hands) What happens as p ! 1? One should expect the limitis the solution to y0 � y = y. . . But the limit isn’t even well-defined for all x. . .

5

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5. (5 points) Consider the IVP (y0 = x+ y2

y(0) = 0.

Let y be a solution to this IVP, and use Euler’s method with step h = 0.1 to approximate y(0.3). Indicateclearly all relevant xk, yk, f(xk, yk). Round your final answer to 3 decimal places.

6

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