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solve every question in details and provide an explanation of how you arrived to the final answer, show steps, write answers in exact form. 7 question total.

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1. (14pts)

2

a. Verify that the given function y = sin ( x + C ) is a solution of the given

differential equation.

dy

= 2x 1− y2 ;

dx

−1 y 1

b. Find an appropriate interval I of definition for the solution in part (a)

[Hint: refer to how the interval of definition was defined]. Justify your answer.

1

2. (14pts) Solve the differential equation by separation of variables

x

dy

= e y+ x ;

dx

x0

2

3. (14pts) Solve the initial value problem for the differential equation

dy

= 1 + y;

dx

y ( 0) = 1

3

4.

(16pts) Given an autonomous first-order DE

dy

= y3 − y

dx

a. Find the critical points

b. Sketch the phase portrait

c. Classify each critical point as an stable (i.e. attractor), unstable (i.e. repeller)

or semi-stable

d. By hand, sketch typical solution curves in the region in the xy-plane

determined by the graphs of the equilibrium solutions.

4

5. (14pts) Solve the differential equation

y3dx + 3xy 2 dy = 0

5

6. (14pts) Verify that the differential equation is Homogeneous, then solve it by

substitution

(4 x − y )dx + (6 y − x)dy = 0

6

7.

(14pts) A tank initially contains 200 gallons of pure water. At time t = 0, a brine solution

of 3 pounds of salt per gallon of water is added to the container at the rate of 4 gallons

per minute, and the well-stirred mixture is drained from the tank at the same rate. Find

the number of pounds of salt in the tank as a function of time.

7

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Tags:

problem solving

differential equation

initial value

separation of variables

homogeneous equation

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