Exam 1 solutions
with(plots):with(DEtools):
is separable with solution y = x + c
is first order linear with integrating factor
which integrates to
condition
so we have
Using the initial
we find C=1 so
plot(x+1)*exp(-(1/2)*x^2),x=-3.3);
1
0
1
2
x
is a Bern
1. Solve y 3y 4y = 2et with ICs y (0) = 1, y (0) = 0 2. Find the general solution to u (x) + 3u (x) + 3u (x) + u(x) = x2 2x + 3 3. A mass of m = 2 kg. stretches a 40 meter spring to 59.6 meters. The constant of viscosity is = 2, the forcing has the form c
Applied Analysis 335 Practice Exam 1b Solutions
February 15, 2004
1. For the ODE
dy dt
= -ty 2
(a) Graph the direction field of the ODE using the points (-1, 0), (0, 0), (1, 0), (-1, 1), (0, 1), (1, 1), (-1, 2), (0, 2), (1, 2) on the graph in figure 1
Fig
335 Practice Exam
September 24, 2008
This practice exam consists is similar in content to a real exam, but longer so that several different types of differential equations can be included. On a real exam, you would have 1.25 hours, but this should take yo
Applied Analysis 335 Practice Exam 1 Answers
1. Solve y (x) 1 y = 6t.
t
R1
Solution: This equation will fall to an integrating factor of = e t dt = 1 . Then we have
t
d
(y 1 ) = 6 y = 6t2 + Ct
dt
t
2. Reproduce the argument that leads to the discovery of
335 Practice Exam
September 24, 2008
This practice exam consists is similar in content to a real exam, but longer so that several different types of differential equations can be included. On a real exam, you would have 1.25 hours, but this should take yo
1
Solve the following problems by separation of variables
1.
dx
dt
= 4t3 x with initial conditions x(0) = 2
2.
dy
dt
= y 1/2 (1 y )
3. Suppose a population of yeast Y (t) grows at a rate proportional to the current population. Write a dierential
equation
The Logistic Equation (a model for population growth) P(t) = r P(t)(1-P(t)/K) Here r=.75 and K=10.
P ' = r P (1 - P/K) 15 r = 0.75 K = 10
10
P 5 0 0 2 4 6 8 10 12 14 16 18 20
We plot a solution curve corresponding to P(0)=5.
1
The Logistic Equation with a
Energy methods and variation.
We frequently can build models by considering the "balance" of kinetic and potential
energy. There is some deep and interesting mathematics (and physics) behind these
methods, but we will only scratch the surface.
The basic i
Calculus of variations and dierential equations of mechanical
systems
John Starrett
New Mexico Institute of Mining and Technology
801 Leroy Place Socorro, NM, 87801
[email protected]
We describe the calculus of variations and use it to obtain the equations
éu.
Math 335 * Ordinary Dierential Equations, Instructor: Rakhfm Aitbayev, 20 September 2012 1
Test 1 "I." Score
------
NAME: 5 0 . . +53%" .
Show all your work for full credit. The textbook or notes are not allowed. You may use
a ca
Calculus of variations and dierential equations of mechanical
systems
John Starrett
New Mexico Institute of Mining and Technology
801 Leroy Place Socorro, NM, 87801
[email protected]
We describe the calculus of variations and use it to obtain the equations
Applied Analysis 335 Practice Exam 2 Solutions
March 27, 2004 1. Solve the linear non-homogeneous equation y + 5y + 6y = -3e-2t + sin(t).
Solution: The characteristic equation is r2 + 5r + 6 = 0 with roots r = -2, -3, so the solutions to the homogeneous p
ODE HW week 10, Fall 2012
November 16, 2012
Solve the following dierential equations by the series method. Assume the
solution is of the form y = n=0 an xn , with derivatives y = n=1 an nxn1 ,
n2
y = n=2 an n(n 1)x
.
1. y y = 0
2. y 2y + y = 0
3. y xy = 0