Math 1B quiz solutions
December 6, 2006
1 Find a general solution to y + 2xy + 2y = 0 as a power series (10 points).
We write y =
n=0
anxn and then take the derivatives:
nanxn1
y=
n=1
y=
n=2
n(n 1)anxn2
Then we substitute these into the dierential equatio
Math 1B quiz solutions
October 18, 2006
1 Find the radius of convergence and the interval of convergence for
xn
.
(3n)!
n=0
(5 points)
Use the ratio test:
lim
n
an+1
xn+1 /(3(n + 1)!
= lim
n
an
xn/(3n)
|x|(3n)!
= lim
n (3n + 3)!
|x|
= lim
n (3n + 1)(3n +
Math 1B quiz solutions
October 20, 2006
1 Does
10n
converge or diverge? (5 points)
n 3n+1
n=1
All the terms are positive, so we can use a limit comparison test with bn =
1/n:
an
10n/(n 3n+1 )
= lim
n bn
n
1/n
lim
1
n 3
= lim
10
3
n
which diverges to becau
Solutions to some limit convergence denition
problems
Dustin Cartwright
October 2, 2006
These are some possible questions which I came up with involving the definition of a convergent series, and solutions to these questions. The solutions
are probably mo
Math 1B Problems
Dustin Cartwright1
1. If f is a continuous function on [a, b] and
b
g(x) =
f (t) dt
x
What is g (x)?
2. Find a function f and a number a such that
x
4+
a
f (t)
dt = 2 x
t2
for all x > 0.
3. Find the value of
n
lim
n
i=1
i3
n4
(Hint: The s
2
x2 + 4 dx.
1. Find
0
e
2. Find
(ln x)2 dx.
1
1 /2
arctan x
dx.
x
0
4. If the curve y = 2x x2 , 0 x 1 is rotated about the x-axis, nd the
area of the resulting surface.
3. Find
(1)n
to within 0.01. (You do not
n+1
n=1
need to carry out the computation, b
Math 1B - Fall 2006
10/09/2006
Integrals
sin2 x cos2 x dx Use 2 sin x cos x = sin 2x, then use
Ex 1.
Ex 2.
0
1
(1 cos 4x) = sin2 2x.
2
4 dx
1
1
dx Try x = tan , dx = sec2 d.
2+1
4x
2
2
Ex 3.
Ex 4.
4 dx
4 + e2x
x
Use Weierstrass substitutions, let t = tan
Math 1B Problems, volume 4
Dustin Cartwright1
1. Solve y 4y = ex cos x using the method of undetermined coecients
with the initial conditions y(0) = 1 and y (0) = 2.
2. Solve y + 3y + 2y = sin(ex ).
3. Solve y 2y + y =
ex
.
1 + x2
4. Find the general solu
Math 1B Problems, volume 3
Dustin Cartwright1
1. Let P (x) = 2 3x2 + 5x3. What is P (n)(0) for n = 1, 2, 3, 4, 5? What is
the Taylor series for P ?
2. Show that the function
J0(x) =
(1)n x2n
22n(n!)2
n=0
satises the dierential equation
x2J0 (x) + xJ0(x) +
Math 1B quiz solutions
October 25, 2006
1
around x = 0. (Hint: Use a
1 + x2
power series you already know) (5 points)
1 Find a power series representation for
This is similar to the expression for the geometric series, so
1
(1)n x2n
=
1 + x2 n=0
2 Show th
Math 1B quiz solutions
October 25, 2006
cn(x 3)n is a power series which converges absolutely for x = 0, what
1 If
n=0
can you say about its radius of convergence? (3 points) What if it converges
conditionally for x = 0? (2 points)
This is a power series
Math 1B quiz solutions
November 29, 2006
1 Find the general solution to y + 2y + y = 5 sin 2x using undetermined coecients (7 points).
The auxillary equation is r 2 + 2r + 1 = 0 which has a root only at r = 1.
Thus the complementary solution is y = ex + x
Math 1B quiz solutions
(Note:
November 29, 2006
tan u du = ln | sec u| + C )
1 Find the general solution to y + 4y + 3y = ex + sin x using the method of
undetermined coecients (7 points).
The auxillary equation is r 2 + 4r + 3 = 0. This has solutions r =
Math 1B quiz solutions
November 15, 2006
1 Find the general solution to
y
x
2
= y + ex . (5 points)
First, we rearrange to get the standard form for a linear dierential equation:
y yx = xex
2
Now we nd the integrating factor:
R
e
x dx
= ex
2
/2
Then we mu
Math 1B quiz solutions
November 14, 2006
1 Find the general solution to the equation y + y/x + x2 1 = 0. (5 points)
This is a linear dierential equation after rearranging to get the standard
linear form:
y + y/x = x2 + 1
First we nd the antiderivative of
Math 1B quiz solutions
November 8, 2006
1 Sketch a direction eld for the dierential equation y = x/y (3 points) and
sketch a solution (2 points). (Note that the dierential equation is not dened
for y = 0).
The solution curve is a half-circle centered at t
Math 1B quiz solutions
November 8, 2006
1 If y = F (x, y) is a dierential equation, y0 = y(x0 ) is an initial condition, and
the step size is h, what is the next point produced by Eulers method? (Hint: the
x coordinate is x0 + h) (5 points)
The point (x0
Math 1B quiz solutions
November 3, 2006
1 Find the power series for
doesnt use
k
n
1
(x+2)2
(2 points) and simplify it to the point that it
, factorials (!), or products of the form 1 2 n (3 points).
We have to use a little bit of algebra to get this into
Math 1B quiz solutions
November 1, 2006
1 Find the power series for
terms (2 points).
3
1 x (3 points). Write out the rst 3 non-zero
Using the binomial series, we get:
3
1 x = (1 x)1/3
=
n=0
=
n=0
1/3
(x)n
n
(1)n
1/3 n
x
n
(1/3)(1/3 1) 2
1/3
x+
x +
1!
2!
Math 1B Problems, volume 2
Dustin Cartwright1
1. Which of the following are p-series or can be treated as p-series? Which
are geometric series? If it is a p-series or geometric series, say whether it
converges.
3 n
(a)
n=0
(b) 1 +
(c)
n=1
1
1
1
+ 3 + 4 +.