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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a. Verify that the given function y = sin ( x + C ) is a solution of the given
= 2x 1− y2 ;
−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.
2. (14pts) Solve the differential equation by separation of variables
= e y+ x ;
3. (14pts) Solve the initial value problem for the differential equation
= 1 + y;
y ( 0) = 1
(16pts) Given an autonomous first-order DE
= y3 − y
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)
d. By hand, sketch typical solution curves in the region in the xy-plane
determined by the graphs of the equilibrium solutions.
5. (14pts) Solve the differential equation
y3dx + 3xy 2 dy = 0
6. (14pts) Verify that the differential equation is Homogeneous, then solve it by
(4 x − y )dx + (6 y − x)dy = 0
(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.
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separation of variables
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