[MA 214, Section 9, Solution to problem 25, section 2.1]
Consider the initial value problem:
ty + 2y =
sin t
,
t
y (/2) = a.
(b) First, we right the equation in the standard form:
sin t
2
y + y = 2 .
t
t
Notice that the coecients of this linear equation a
[MA 214, Section 9, Solution to problem 5, section 2.2]
Solve the dierential equation
y = (cos2 x)(cos2 2y ).
Solution: This is separable equation. Separating the variables, assuming cos 2y = 0,
gives:
dy
= cos2 x dx.
cos2 2y
Integrating the left-hand sid
[MA 214, Section 9, Solution to problem 4, section 2.3]
A tank with a capacity of 500 gal originally contains 200 gal of water with 100 lb of salt
in solution. Water containing 1 lb of salt per gallon is entering at a rate of 3 gal/min,
and the mixture is
[MA 214 009: Worksheet 2, March 22, 2013]
1 Solve the equation
y + y = sec3 t,
(0 < t < )
Solution: The characteristic roots of the equation are i, so the general solution
of the homogeneous equation is
yp = C1 cos t + C2 sin t.
To nd a particular solutio
[MA 214, Section 9 First Midterm Exam Solutions]
The exam has 4 questions plus a bonus question. Please write your solutions on the
provided paper and use this page as a cover sheet. Write your name on top of each
page. No textbook, no notes. Calculators
[MA 214, Section 9: Nine rst order equations. Solutions]
For each of the following equations,
(a) Determine the appropriate method of solution.
(b) Solve the equation.
Part (a) should be done today in class. Part (b) should be a good practice before
the r
[MA 214, Section 9: Five Problems on Second Order Equations Solutions]
None of the following problems is too dicult to be on your Fridays
exam. Taken together, they would make a challenging exam. Solutions
will be posted by tomorrow 12 PM.
1 For which val
[MA 214, Section 9: Seven problems. Solutions]
1 Let y1 , y2 be two solutions of an equation of the form
y + p(t)y + q (t)y = 0
with continuous p(t), q (t), and let W (y1 , y2 )(t) be the Wronskian of y1 , y2 . Prove that
W (y1 , y2 ) =
y1 y2
.
y1 y2
Solu
[MA 214, Section 9, Quiz 1, Solution]
Consider the following initial value problem:
y = 2y 1,
y (0) = y0 .
(a) Solve this initial value problem.
(b) For the solution in (a), compute y (ln 2). Observe that y (ln 2) depends linearly
on y0 .
(c) If t1 > 0 is
[MA 214, Section 9, Quiz 2, Solution]
Consider the following initial value problem:
y y = 1 + 3 sin t,
y (0) = y0 .
(a) Find the general solution of the equation.
(b) Find all the values of y0 for which the solution of the initial value problem remains
bo
[MA 214, Section 9, Quiz 3, Solution]
Consider the following initial value problem:
2x
y =
,
y (0) = 1.
3 + 2y
(a) Find the solution y (x) in explicit form (i.e. solve for y ).
(b) Find all the values of x for which the integral curve y = y (x) intersects
[MA 214, Section 9, Quiz 4, Solution]
(a) Solve the initial value problem
4y + 13y + 9y = 0,
y (0) = 4,
y (0) = 1.
(b) For the solution y (t) found in (a), nd
lim et y (t).
t+
Solution: (a) The characteristic equation
42 + 13 + 9 = 0
has roots
=
13
169 4
[MA 214, Section 9, Quiz 5, February 27, 2013]
1 Find the general solution of
y y = 2et t2 .
Solution: The characteristic roots are = 1. Thus the general solution of the
homogeneous equation
y y = 0
is
y = C1 et + C2 et .
For the particular solution, we r
[MA 214, Section 9, Quiz 7, Solutions]
For each of the following functions, nd the Laplace transform. You can, if you wish,
use the provided table of Laplace transforms.
1
f (t) = (t + 2)2 .
Solution: f (t) = t2 + 4t + 4, so
F (s) =
2
4
4
+ 2+ .
3
s
s
s
2