Quiz 1 Solutions - differential equations PDF

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differential equations...

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UNIVERSITY OF REGINA Department of Mathematics and Statistics Differential Equations I Mathematics 217 (Section 001) Fall 2019 Instructor: Liviu Mare Quiz # 1 (September 20th ) Solutions

Time: 20 min.

Full Name:

Pages: 3

Student Number:

INSTRUCTIONS 1. The left side pages are to be used as scrap paper. They are provided for rough work and checking only and will not be graded unless you expressly indicate there is work to be found there. 2. The marks allocated for each question are found to the left of the question.

Good Luck!

For instructor use only: Page:

2

3

Total

Marks:

6

4

10

Score: Marks

MATH 217 (3)

Page 2

Quiz # 1

1. Determine all possible integers m with the property that y emx is a solution of the differential equation y y 6y 0.

Solution. Substitute y

emx into the equation. Take into account that y y

memx m2 emx .

Obtain: m2 emx memx 6emx m2 m 6 0 m 2 m 3 0 m 2 or m 3.

(3)

0 divide by emx

2. Consider the differential equation dy dx

y 2 2.

(a) Find all critical points of the equation. (b) Draw the corresponding phase portrait. (c) Classify each critical point as asymptotically stable, unstable, or semistable.

Solution. (a) Solve the equation y point is 2.

2

2

0. The only solution is y 2. So the only critical

(b)

(c) The critical point 2 is semistable.

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MATH 217 (4)

Page 3

Quiz # 1

3. Find the general solution of the differential equation dy dx

x . y2

(Do not search for constant solutions, just find the general solution.)

Solution.

dy x dx y 2 dy y2 x dx y 2 dy xdx y 2 dy 1 3 y 3 y3 y Denote c

xdx

1 2 x 2

c1

3 2 x 3c1 2 3 2 3 x 3c1 2

3c1 and get the final answer: y

3

3 2 x 2

c.

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