math problem
sa3p14
Math.pdf
2. (8 points ) True or False? Circle one. No justifications necessary!
(a ) T or F: The function y (t ) = sin (5t ) is a solution t o the differential equation dy j dt = 5y.
(b ) T or F: Every autonomous differential equation is separable.
(c) . T or F: Every separable different ial equation is linear.
(d ) T or F: It is possible for an autonomous equation dy j dt = f (y ) to only have two equilibrium points that are both node.
3. (8 points ) Circle all that apply. (Note: "Homogeneous" only applies to linear equations. )
(a ) dy
Separable Autonomous Linear Homogenous - -y = t dt
(b ) dy ') ?
Separable Autonomous Linear Homogenous - = 2ty· + 3t-y dt
(c) dy ty2 + t
Separable Autonomous Linear Homogenous - dt y
(d ) dy dt = y (y + 2)2 Separable Autonomous Linear Homogenous
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Other
Other
Other
Other
4. (10 points) Find the general solution. dy ty
dt 2 + t 2
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5. (12 p oints ) CHOOSE ONE (AND ONLY ON E ) OF THE FOLLOWING QU ESTIONS !
Solve the following IVP:
(i) ~~ = - 3y + 3 sin( 4t), y(O) = 2
(ii) ~~ = - 6y + 5e- 6t, y(O ) = 2 I'm going to choose: _ _
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6. (10 points ) Find the general solution.
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7. The following is the sketch of the graphs of several solutions to an autonomous differential equation dy j dt = J (y ).
(a) ( 5 points ) Sketch the phase line for the difierential equation dy / dt = J (y ). Also , identify the equilibrium points as sinks, sources , or nodes.
y
r-1 ~ I
(b) (6 points ) Give a rough sketch of the graph of the corresponding funct ion J( y ). Clearly label the axes.
(c) (3 points ) Give a possible formula fo r the function f( y ) whose graph agrees (qualitativel y) with the rough sketch in part (b ) .
J (y) = ------ ---- ----
5
=
8. (3 points ) Determine the differential equation that corresponds to each slope field, and state briefly how you know your choice is correct.
(a) dy dt = J (y ) (b) ~~ = j (t ) y
\ - ,. " I \ - ,. " 1: \ - / " I, \ - ,. " I - / " 1: - / " 1: - / " - / " - / " I' - / " ,,
y
' ' ' ' ' ' ' ' ' t ' ' ' ' : t - - - - - - " " " " " " " " " - / / i: - / " I - / " I' / / I I / / / / I / ,. / / / / / I " / / / / / / / " - ,. "
i= - / " - ,. "
- - - ' ' ' ' ' ' ·' ' ' I I I I I \ I I I
(i) ( ii )
9. ( 4 points ) Given an example of a first-order autonomous differential equation whose phase line loo ks like:
dy
dt
6
6
-2
Part II: CALCULATOR ALLOWED.
10. (20 points) Consider a large tank containing sugar water that is to be made into soft drinks. Sup- pose :
• The tank initially contains 150 m3 of sugar water with 3 kg sugar. Moreover , the amount flowing in is the same as the amount fl owing out , so there are always 150m3 in the tank.
• The tank is kept well mixed , so the sugar concentration is uniform throughout the tank.
• Sugar water containing 7 kg / m 3 enters the t ank through pipe A at a rate of 3 m 3 / min.
• Sugar water cont aining 11 kg/ m 3 enters the tank through pipe B at a rate of 2 m 3 / min.
• Sugar water leaves the tank through pipe C at a rate of 5m 3 / min.
Model this using a differential equation. That is , state the IVP. Solve the IVP you came up with in order to determine the amount of sugar in the vat after 10 minutes . You may use the next page if you need more space!
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ll. (10 points ) Use Euler 's Method wi th 6.t = 0.5 to approximate y(1.5) of the followi ng IVP:
dy = 2y - 3t3 0 (0 ) 0 ') dt ' y = . .).
Round your answer to 3 decimal places. You may complete the foll owing table.
kl tk I I Yk
0 0 I 0.3 1 I
I
2 1 I I
3[ ! ***********
y(1.5 ) ;:::; -------
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