Victor Camocho
math2250fall2011-2
WeBWorK assignment number Homework 2 is due : 09/01/2011 at 11:00pm MDT.
The
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This file is /conf/snippets/setHeader.pg you can use it as a model for creating files which introduce each problem set.
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C. Consider the curve in the plane defined by setting y0 = 0
– this should correspond to the points in the picture
where the slope is zero.
D. Setting y0 equal to a constant other than zero gives the
curve of points where the slope is that constant. These
are called isoclines, and can be used to construct the
direction field picture by hand.
1. (1 pt) hw2/p1.pg
For each initial value problem below, determine whether or not a
solution exists, and if so, determine whether or not that solution
is unique within a rectangular interval about the initial point. If
a solution does not exist enter ”N”. If a solution exists but is not
guarunteed to be unique, enter ”E”. If a solution exists and is
unique, enter ”U”.
1.
y0 (x) = 2x−2 y5 , y(0) = −1.
2.
y0 (x) =
√
5 y, y(0) = 0.
3.
y0 (x) =
p
4.
y0 (x) =
x−1
y ,
5.
y0 (x) = y ln(1 + y3 ), y(0) = 0.
6.
y0 (x) = x2 − y2 , y(−1) = 7.
1.
2.
3.
4.
x2 − y, y(2) = 1.
2
y0 = 2xy + 2xe−x
y0 = y + xe−x + 1
y0 = 2 sin(3x) + 1 + y
y
y0 = + 3 cos(2x)
x
y(0) = 5.
2. (1 pt) Library/Rochester/setDiffEQ2DirectionFields/ur de 2 3.pg
Match the following equations with their direction field. Clicking on each picture will give you an enlarged view. While you
can probably solve this problem by guessing, it is useful to try
to predict characteristics of the direction field and then match
them to the picture.
Here are some handy characteristics to start with – you will
develop more as you practice.
A
A. Set y equal to zero and look at how the derivative behaves along the x-axis.
B. Do the same for the y-axis by setting x equal to 0
1
B
8. (1 pt) Library/Rochester/setDiffEQ1/ur de 1 4.pg
Find the value of k for which the constant function x(t) = k is a
dx
solution of the differential equation 8t 5 − 8x − 9 = 0.
dt
9. (1 pt) Library/ma123DB/set6/s9 3 8.pg
Solve the separable differential equation for u.
du
= e2u+7t
dt
C
D
Use the following initial condition: u(0) = 2
3. (1 pt) Library/274/Lin1stord/prob10.pg
Find the general solution to the differential equation
x2 + 1xy + x
u=
dy
=0
dx
10. (1 pt) Library/Rochester/setDiffEQ3Separable/ns7 4 8b.pg
Find the function y = y(x) (for x > 0 ) which satisfies the separable differential equation
Find the integrating factor, u(x) =
.
dy 7 + 18x
=
; x>0
dx
xy2
.
Find y(x) =
Use C as the unknown constant.
with the initial condition y(1) = 4.
.
y=
4. (1 pt) Library/274/Lin1stord/prob12.pg
Solve the initial value problem
11. (1 pt) Library/Rochester/setDiffEQ3Separable/ur de 3 6.pg
A. Find y in terms of x if
dy
− y = 7et + 8e9t
dt
with y(0) = 8.
y=
dy
= x2 y−5
dx
and y(0) = 5.
.
y(x) =
5. (1 pt) Library/274/Lin1stord/prob4.pg
Solve the following initial value problem:
t
B. For what x-interval is the solution defined?
(Your answers should be numbers or plus or minus infinity. For
plus infinity enter ”PINF”; for minus infinity enter ”MINF”.)
dy
+ 9y = 1t
dt
with y(1) = 7.
Find the integrating factor, u(t) =
and then find y(t) =
The solution is defined on the interval:
<x<
.
,
12. (1 pt) Library/Rochester/setDiffEQ3Separable/osu de 3 11.pg
Find the particular solution of the differential equation
.
6. (1 pt) Library/Rochester/setDiffEQ4Linear1stOrder/ur de 4 3.pg
Find the function satisfying the differential equation
dy
= (x − 4)e−2y
dx
satisfying the initial condition y(4) = ln(4).
Answer: y =
.
Your answer should be a function of x.
y0 − 5y = 6e8t
and y(0) = 2.
y=
.
.
13. (1 pt) Library/Rochester/setDiffEQ3Separable/osu de 3 12.pg
Find the particular solution of the differential equation
7. (1 pt) Library/Rochester/setDiffEQ4Linear1stOrder/ur de 4 14.pg
Find the function y(t) that satisfies the differential equation
x2 dy
1
=
y2 − 6 dx 2y
√
satisfying the initial condition y(1) = 7.
Answer: y =
.
Your answer should be a function of x.
2
dy
− 2ty = 12t 2 et
dt
and the condition y(0) = −2.
y(t) =
.
2
15. (1 pt) hw2/p3.pg
A tank initially contains 60 gallons of pure water. Brine containing 2 pound of salt per gallon enters the tank at 4 gal/min,
while the perfectly mixed solution leaves the tank at 6 gal/min.
Note that the tank is completely empty after exactly 30 minutes.
In this problem, we will let V (t) represent the volume of solution (in gallons) in the tank at time t, x(t) represent the pounds
of salt in the tank at time t, and c(t) represent the concentration
of the solution in the tank at time t.
14. (1 pt) hw2/p2.pg
The intensity of light, I, at a depth x meters below the surface of
a lake satisfies the following differential equation:
dI
= −0.3I
dx
The intensity at the surface is referred to as I0 .
(1) At what depth is the intensity of light 25% of the intensity
meters.
at the surface?
(1) Find the amount of salt in the tank after t minutes. That
is, determine the function x(t).
(2) What is the intensity 6 meters below the surface as a
fraction of the surface intensity?
×I0 .
x(t) =
(3) At what depth will the intensity be 1% of the intensity at
the surface?
meters.
(2) What is the maximum amount of salt in the tank during
this 30 minute period?
pounds.
c
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