Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Main points of this exam paper are: Solution, Equation, Satisfies, Verify, Implicit, Slope, Determines, Slope Decreases, Artificial Snow, Rate
Typology: Exams
1 / 10
This page cannot be seen from the preview
Don't miss anything!
There are 9 questions on this exam. Question 9 is worth 12 points. Each other question is worth 11 points. The points assigned to each part of the question are indicated at the start of the part.
Show all your work. Partial credit may be given.
The use of calculators, books, or notes is not permitted on this exam.
Please turn off your cell phone before starting this exam.
Time limit 1 hour and 15 minutes.
ty′^ + 2y = 2 t > 0
This is a linear equation. 2pts It must be put into the following form:
y′^ +
t
y =
t
3pts p =^2 t
t dt^ = 2 ln^ t^ = ln^ t
2
The integrating factor is μ = e
p dt (^) = t 2
3pts Multiplying through by the integrating factor gives (yt^2 )′^ = 2t
1pts Integrating gives yt^2 = t^2 + C
Finally y = 1 + Ct−^2 (Do not remove points of leaving in implicit form)
b. 2pts Find the solution of the above equation which satisfies y(1) = 0. 2pts Plug in t = 1 and y = 0 to find C = − 1
4pts We see that y′′^ = dv dt = dv dydy dt = dv dy v
The ODE is now dv dy v = yv^3
2pts y =const solves the equation. So we assume that v is not zero (the function) and the ODE is now v−^2 dv dy = y
2pts ∫ We integrate both sides with respect to y: v−^2 dv dy dy =
y dy
−v−^1 =^12 y^2 + C 3pts We now return to viewing t as the independent variable: −2 = (y^2 + C)y′^ This is easy to integrate with respect to t: − 2 t + D =^13 y^3 + Cy Take off 1pt for omitting either one of C or D and 2pts for omitting both. Take off 1pt for not mentioning the solution y=const.
b. 5pts Suppose that after 10 days of operation there are 5x10^6 cubic meters of snow on the slopes. Use Euler’s method with one step to approximate the amount of snow on 11th day. 1pts The choice of t 0 does not influence the answer to this question since the ODE is autonomous. However, for the sake of definiteness take t 0 = 10. Then V 0 = 5 × 106 and h = 1.
2pts Now V ′(t 0 , V 0 ) = − 10 15 × 106 + 10^6 =^12 × 106
2pts Therefore V 1 = V 0 + V ′(t 0 , V 0 )h =^112 × 106
ANS. 2pts The characteristic polynomial r^2 + 6r + 10 has complex roots r = − 3 ± i 2pts Therefore y = et(c 1 cos t + c 2 sin t).
b. 3pts Find the general solution of y′′^ + 6y′^ + 9y = 0
ANS. 1pts The characteristic polynomial r^2 + 6r + 9 has double root r = − 3 2pts Therefore y = e−^3 t(c 1 + c 2 t).
c. 2pts Solve the initial value problem:
y′′^ + 6y′^ + 9y = 0 y(0) = 1, y′(0) = 0
2pts Setting t = 0 in the above gives c 1 = 1 and c 2 = 3. Therefore y′^ = − 3 y + e−^3 tc 2.
d. 2pts Find limt→∞ y(t), where y(t) is the solution found in Part c. 2pts By L’Hospital’s rule, the limit is zero.
a. 3pts Determine all the equilibrium solutions of this ODE.
−y^2 + 4y − 3 = −(y^2 − 4 y + 3) = −(y − 3)(y − 1)
y = 3 and y = 1 are the equilibrium solutions.
b. 4pts Sketch a direction field for this ODE. Use the Direction Field applet to sketch the direction field.
c. 4pts For each equilibrium solution found in Part a. determine whether it is asymptotically stable or unstable. 2pts y = 3 is asymptotically stable.
2pts y = 1 is unstable
ANS. 1pts Rewrite equation y′^ + (^) t −^3 1 y = (^) t −^2 . 2pts This is a linear ODE. The coefficients are discontinuous at t 0 = 1. So uniqueness is NOT guaranteed when for all (t 0 , y 0 ) with t 0 = 1.
b. 3pts Consider the ODE (t−1)y′^ +y^1 /^3 = 2. Determine all pairs (t 0 , y 0 ) for which the uniqueness of the solution to the IVP y(t 0 ) = y 0 is NOT guaranteed?
ANS. 1pts Rewrite equation y′^ = f (t, y) = − (^) t −^1 1 y^1 /^3 + (^) t −^2 . This is a nonlinear ODE. The f (t, y) discontinuous at t 0 = 1. The partial with respect to y, fy is discontinuous if t 0 = 1 or if y 0 = 0. 2pts So uniqueness is NOT guaranteed when for all (t 0 , y 0 ) with t 0 = 1 or y 0 = 1.
c. 3pts Which one of the following is a second order ODE? Circle it.
y^2 = cos^2 t y(y′)^2 = sin t y′′ y
= tan t
3pts The last equation is second order
d. 3pts Which one of the following is a linear ODE? Circle it.
t^4 y′^ + t^3 y = sin^2 t 1 = y y′^ t = y′ey
3pts The first equation is linear.
