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Exam 3 in MAT 353: Differential Equations (Spring 2000) - Prof. Kirk Jones, Exams of Differential Equations

The third exam for the mat 353: differential equations course taught by professor jones in spring 2000. The exam covers various topics related to solving ordinary differential equations (odes), including euler's method and heun's method. Students are required to solve problems involving finding solutions, integrating factors, and error estimation.

Typology: Exams

Pre 2010

Uploaded on 08/16/2009

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MAT 353
(Jones Spring 2000)
Differential Equations
Exam 3 (4.4 - Eulers Method)
Write your SS # on each page you turn in to be graded - do not write your name anywhere on the
exam. Solve each of the following problem showing all work for partial credit.
1.
Consider the ODE
where
. Show that
if
,
and
solves the
given second order ODE for all
.
2.
Consider the ODE
where
. Show that
if
, and
, show that
.
3.
Solve the following linear ODEs.
a.
b.
c.
4.
Solve the following linear homogeneous ODEs.
a.
b.
c.
d.
5.
Find a particular solution
for each ODE below.
a.
b.
c.
pf2

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MAT 353 (Jones Spring 2000) Differential Equations Exam 3 (4.4 - Euler’s Method) Write your SS # on each page you turn in to be graded - do not write your name anywhere on the exam. Solve each of the following problem showing all work for partial credit.

  1. Consider the ODE where. Show that if , and , then solves the given second order ODE for all.
  2. Consider the ODE where. Show that if , and , show that.
  3. Solve the following linear ODEs. a. b. c.
  4. Solve the following linear homogeneous ODEs. a. b. c. d.
  5. Find a particular solution for each ODE below. a. b. c.
  1. Consider on (0, ). a. Show that is a solution of. b. Show that the substitution transforms into . c. Show that the substitution transforms into . d. The ODE is a linear ODE with integrating factor. Solve the ODE. e. Using 6d and the above substitutions, show that the general of is given by.
  2. a. Use Euler’s method to estimate the solution of ; at t = 1 with a step size of h = 0.5. b. Use Heun’s method to estimate the solution of ; at t = 1 with a step size of h = 0.5. c. Solve the IVP ; d. Find the error in computing y (1) via Euler’s and Heun’s methods. e. Estimate the error in computing y (1) under Euler’s and Heun’ method if h = 0.125. f. Using the fact that , explain why Euler’s method will always under estimate the actual solution to this particular IVP while Heun’s method will always over estimate the solution.