MATH 285 Grossmont College Differential Equation Problem Solving Discussion

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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
x0
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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