Math 285 Mid Term 1 Practice Sp, 2017 Manfroi
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8. Determine the integrating factor for the
following differential equation. t2 y' ˗ y t2 sin t = t
cost t Answer: Integrating Factor = e cos t |
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9. a. Find
the general solution to the initial value problem: dy/dt = [ (3y + 4) / (4t + 5) ]2 y(˗ 1) =
˗ 1 answer: (1/3) (3y + 4)˗1
= (1/4) (4t + 5)˗1 + C b.
Find the unique solution satisfying the initial condition . (You don't need to solve for y.) answer; (1/3) (3y + 4)˗1 = (1/4) (4t + 5)˗1 +
1/12 c.
On what interval does the solution exist? |
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10. Consider the initial value problem: y'' ˗ 4 y' + 4 y
= 0,
y(0) = 2a, y'(0) = b a.
Find a fundamental set of solutions for the differential equation. Answer:
y1(x) = e
2x , y2(x) = x e 2x b. Find the general solution of the
differential equation. Answer:
y(x) = C1e 2x + C2
x e 2x c.
Find the solution of the initial value problem. Answer: y(x)
= a e 2x + (
b ˗ 2a) x e 2x |
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11. A certain species has a population
level p(t) modeled by the initial value problem: dp/dt = ˗ 3p(2
˗ p)(8 ˗ p), p(0) = po > 0
a.
Draw the phase diagram and label all critical values/equilibrium
solutions for this model and the direction of flow
between them. Answer:
Critical values at p = 0, 2,
and 8; 0 and 8 are stable; 2 is unstable b.
On the p(t) vs t graph draw some of the
solution curves, including the equilibrium solutions and some solution curves
above and below each of them. c.
Under what conditions on po will
the population go extinct? When will
it happen? Answer:
Population goes extinct if it starts out less than 2 after a very long
time. |