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Implicit Solution - Differential Equations - Exam, Exams of Differential Equations

Some keywords in Differential Equations are Convolution, Laplace Transform, Implicit Solution, Initial Condition, Integrating Factor, Autonomous Differential Equation, Appropriate Substitution. Some points of this exam paper are: Implicit Solution, Solutions, Initial Condition, Differential Equation, Initial Value Problem, Acceptable, Population

Typology: Exams

2012/2013

Uploaded on 03/31/2013

parvani
parvani 🇮🇳

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MA 238-01
Ch. 1-4, 6 Final Exam score
Name:
1 August 2001
Note: Be sure to show all of your work in order to receive credit for a correct answer.
1. Find all solutions for the given differential equation and then find the solution that satisfies
the initial condition y(0)=0. (9 points)
dy
dt +2ty =t
2. Find the solution (an implicit solution is acceptable) for the initial value problem (8 points)
dy
dt =ycos t
1+2y2,y(0)=1
3. Let y(t) denote the size of a population, say in thousands, of creatures, and suppose that y
satisfies the differential equation
dy
dt =−y2+10y−16
Determine if there are any equilibrium solutions. Discuss the features of this system (use
words like harvesting,stable,carrying capacity, etc.). Sketch a representative collection of
solution curves on the axes provided. (8 points)
t
y
pf3
pf4
pf5

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MA 238-

Ch. 1-4, 6

Final Exam

score

Name:

1 August 2001

Note: Be sure to show all of your work in order to receive credit for a correct answer.

  1. Find all solutions for the given differential equation and then find the solution that satisfies the initial condition y(0) = 0. (9 points)

dy dt

  • 2ty = t
  1. Find the solution (an implicit solution is acceptable) for the initial value problem (8 points)

dy dt

y cos t 1 + 2y^2

, y(0) = 1

  1. Let y(t) denote the size of a population, say in thousands, of creatures, and suppose that y satisfies the differential equation

dy dt

= −y^2 + 10y − 16

Determine if there are any equilibrium solutions. Discuss the features of this system (use words like harvesting, stable, carrying capacity, etc.). Sketch a representative collection of solution curves on the axes provided. (8 points)

t

y

  1. Use the method of Picard to find approximations to the solution of the differential equation y′^ = t^2 y − t, y(0) = 0 by computing the Picard iterates y 0 , y 1 , and y 2. (9 points)
  2. For the differential equation dy dt

= y − t^2 with initial condition y(0) = 2, use Euler’s method to approximate the value of y(1) using 2steps (i.e., use h = 0.5). Then use the Runge-Kutta method to estimate y(1) using just one step (i.e., use h = 1). (10 points)

Euler Method

Runge-Kutta Method

  1. A long spring that is suspended from a ceiling. If a 2kilogram mass is attached to the spring, it stretches the spring 1 meter (i.e., the new equilibrium position is 1 meter lower). If the mass is pulled down an additional 2meters and released from rest, find the equation of motion for the mass (assuming no damping). What damping should be used in this system in order to produce a critically damped system? Sketch the graphs of the undamped and critically damped solutions on the axes provided. (9 points)

t

y

  1. A pendulum is suspended from a pivot as discussed in class. The variable θ measures the angle (in radians) of the pendulum arm from the downward vertical position. The graph shows a plot of θ versus t for a damped, true (non-linearized) pendulum. Estimate the initial conditions. Describe the motion of the pendulum based on what you see in the graph. Estimate the value of t when the pendulum bob first passes over the top (its highest position) and also when the bob first passes through its lowest position. On the empty set of axes, sketch the phase plane graph of θ′(t) versus θ(t) for the graph on the left. (10 points) - -

0

2

4

6

8

10

12

14

theta(t)

(^5 10 15) t 20 25 30

θ(t)

θ′(t)

  1. Find the functions f (t) whose Laplace transforms are as follows: (5 points each)

(a) Lf =

s^2 − 1

(b) Lf = se−^3 s s^2 + 9

  1. Find the Laplace transform of the solution of the IVP

2 y′′^ + y′^ + 2y = h(t) y(0) = 0 y′(0) = 0

where h(t) = u 5 (t) − u 20 (t) = step(t − 5) − step(t − 20). Note: You do not need to solve this IVP, you just need to find the Laplace transform. (9 points) Note: You do not need to solve this IVP, you just need to find the Laplace transform. (8 points)