Answers
1. The given equation is probably supposed to read
y'' - 2y' - 3y = 64x exp(-x)
First consider the homogeneous equation,
y'' - 2y' - 3y = 0
which has characteristic equation
r² - 2r - 3 = (r - 3) (r + 1) = 0
with roots r = 3 and r = -1. Then the characteristic solution is
[tex]y = C_1 e^{3x} + C_2 e^{-x}[/tex]
and we let y₁ = exp(3x) and y₂ = exp(-x), our fundamental solutions.
Now we use variation of parameters, which gives a particular solution of the form
[tex]y_p = u_1y_1 + u_2y_2[/tex]
where
[tex]\displaystyle u_1 = -\int \frac{64xe^{-x}y_2}{W(y_1,y_2)} \, dx[/tex]
[tex]\displaystyle u_2 = \int \frac{64xe^{-x}y_1}{W(y_1,y_2)} \, dx[/tex]
and W(y₁, y₂) is the Wronskian determinant of the two fundamental solutions. This is
[tex]W(y_1,y_2) = \begin{vmatrix}e^{3x} & e^{-x} \\ (e^{3x})' & (e^{-x})'\end{vmatrix} = \begin{vmatrix}e^{3x} & e^{-x} \\ 3e^{3x} & -e^{-x}\end{vmatrix} = -e^{2x} - 3e^{2x} = -4e^{2x}[/tex]
Then we find
[tex]\displaystyle u_1 = -\int \frac{64xe^{-x} \cdot e^{-x}}{-4e^{2x}} \, dx = 16 \int xe^{-4x} \, dx = -(4x + 1) e^{-4x}[/tex]
[tex]\displaystyle u_2 = \int \frac{64xe^{-x} \cdot e^{3x}}{-4e^{2x}} \, dx = -16 \int x \, dx = -8x^2[/tex]
so it follows that the particular solution is
[tex]y_p = -(4x+1)e^{-4x} \cdot e^{3x} - 8x^2\cdot e^{-x} = -(8x^2+4x+1)e^{-x}[/tex]
and so the general solution is
[tex]\boxed{y(x) = C_1 e^{3x} + C_2e^{-x} - (8x^2+4x+1) e^{-x}}[/tex]
2. I'll again assume there's typo in the equation, and that it should read
y''' - 6y'' + 11y' - 6y = 2x exp(-x)
Again, we consider the homogeneous equation,
y''' - 6y'' + 11y' - 6y = 0
and observe that the characteristic polynomial,
r³ - 6r² + 11r - 6
has coefficients that sum to 1 - 6 + 11 - 6 = 0, which immediately tells us that r = 1 is a root. Polynomial division and subsequent factoring yields
r³ - 6r² + 11r - 6 = (r - 1) (r² - 5r + 6) = (r - 1) (r - 2) (r - 3)
and from this we see the characteristic solution is
[tex]y_c = C_1 e^x + C_2 e^{2x} + C_3 e^{3x}[/tex]
For the particular solution, I'll use undetermined coefficients. We look for a solution of the form
[tex]y_p = (ax+b)e^{-x}[/tex]
whose first three derivatives are
[tex]{y_p}' = ae^{-x} - (ax+b)e^{-x} = (-ax+a-b)e^{-x}[/tex]
[tex]{y_p}'' = -ae^{-x} - (-ax+a-b)e^{-x} = (ax-2a+b)e^{-x}[/tex]
[tex]{y_p}''' = ae^{-x} - (ax-2a+b)e^{-x} = (-ax+3a-b)e^{-x}[/tex]
Substituting these into the equation gives
[tex](-ax+3a-b)e^{-x} - 6(ax-2a+b)e^{-x} + 11(-ax+a-b)e^{-x} - 6(ax+b)e^{-x} = 2xe^{-x}[/tex]
[tex](-ax+3a-b) - 6(ax-2a+b) + 11(-ax+a-b) - 6(ax+b) = 2x[/tex]
[tex]-24ax+26a-24b = 2x[/tex]
It follows that -24a = 2 and 26a - 24b = 0, so that a = -1/12 = -12/144 and b = -13/144, so the particular solution is
[tex]y_p = -\dfrac{12x+13}{144}e^{-x}[/tex]
and the general solution is
[tex]\boxed{y = C_1 e^x + C_2 e^{2x} + C_3 e^{3x} - \dfrac{12x+13}{144} e^{-x}}[/tex]
Related Questions
An inventor makes a rectangular array of 1024 LEDs. Each row will
have 16 LEDs. How many rows of LEDs does he need?
Answers
Answer:
64
Step-by-step explanation:
Since the total area of the rectangle (in this case number of leds) is:
Total LEDs = rows * columns
you can just replace the values:
1024 = 16 * x
And then divide 1024 / 16 to find x
x = 64
Let g(x)=Intragal from 0 to x f(t) dt, where r is the function whos graph is shown.
Picture attached with question and graphs. As well as points needed
Answers
If
[tex]\displaystyle g(x) = \int_0^x f(t) \, dt[/tex]
then g(x) gives the signed area under f(x) over a given interval starting at 0.
In particular,
[tex]\displaystyle g(0) = \int_0^0 f(t) \, dt = 0[/tex]
since the integral of any function over a single point is zero;
[tex]\displaystyle g(4) = \int_0^4 f(t) \, dt = 8[/tex]
since the area under f(x) over the interval [0, 4] is a right triangle with length and height 4, hence area 1/2 • 4 • 4 = 8;
[tex]\displaystyle g(8) = \int_0^8 f(t) \, dt = 0[/tex]
since the area over [4, 8] is the same as the area over [0, 4], but on the opposite side of the t-axis;
[tex]\displaystyle g(12) = \int_0^{12} f(t) \, dt = -8[/tex]
since the area over [8, 12] is the same as over [4, 8], but doesn't get canceled;
[tex]\displaystyle g(16) = \int_0^{16} f(t) \, dt = 0[/tex]
since the area over [12, 16] is the same as over [0, 4], and all together these four triangle areas cancel to zero;
[tex]\displaystyle g(20) = \int_0^{20} f(t) \, dt = 24[/tex]
since the area over [16, 20] is a trapezoid with "bases" 4 and 8, and "height" 4, hence area (4 + 8)/2 • 4 = 24;
[tex]\displaystyle g(24) = \int_0^{24} f(t) \, dt = 64[/tex]
since the area over [20, 24] is yet another trapezoid, but with bases 8 and 12, and height 4, hence area (8 + 12)/2 • 4 = 40, which we add to the previous area.
A shirts costs $20, the cost of the shirt increases 30%, what is the price of the shirt now?
Answers
Answer: 10
Step-by-step explanation:
HELP!!!! ME PLSSSSSSSSSSSSSSSSSSSSS
Answers
Answer:
A for the Venn Diagram and B for the chart.
Step-by-step explanation:
1. Domain: (-2, -1, 0, 2)
Range: (-4, -2, 2)
2. Domain: (-4, 2, 1, 4)
Range: (-2, 1, 4, 4)
The domain represents the values of the independent variable (x) and the range represents the values of the dependent variable (y). The values of the independent variable are inputs that result in an output in a relation. The values in the range are outputs that correspond to inputs from the independent variable.
A desk is 39 inches wide. What is the width in yards
Answers
Answer:
1.0833 yd
Step-by-step explanation:
Inches Yards
37" 1.0278 yd
38" 1.0556 yd
39" 1.0833 yd
40" 1.1111 yd
Have a great day! :D
Compare -8 and 3. Which statement is not true?
Answers
Answer:
B
Step-by-step explanation:
it is the opposite of A, which is true, as |-8| = 8
13. A stack of lumber contains the following: three 1" x 6" X 8' pieces of yellow pine. six 2" x 10" x 12' pieces of redwood. six 1" x 8" 10' pieces of fir. How many total board feet of lumber are in this stack? *
180 board ft
162 board ft
192 board ft
172 board ft
Answers
Answer:
Step-by-step explanation:
A board foot is 1 inch thick by 1 ft by 1 ft
(3) 1 x 6 8' are already 1 inch thick ½ ft wide and 8 ft long
3 x 1 x (6/12) x 8 = 12 board ft.
(6) 2" x 10" x 12' are 2 thick x 10/12 ft wide by 12 ft long
6 x 2 x (10/12) x 12 = 120 board ft.
(6) 1" x 8" x 10'
6 x 1 x (8/12) x 10 = 40 board ft
12 + 120 + 40 = 172 board ft
Write in order least to greatest!
Answers
Hello! :)
√99=9.95...
π²=9.9
9.8888...
So I think the numbers should be written like this:
9.8888, √99, π²
Hope it helps!
~Just a determined gal
[tex]MysteriousNature[/tex]
So There Are 3 Dogs In 25 Parks How Many Dogs In Parks Are There So This Is 3×25=? I mean It's Ez
Answers
Answer: You are right! Just multiply for your answer!
Step-by-step explanation:
HELP ME PLEASE ASAP ASAP
Answers
which graph be represents the soulution set of y[tex]\leq[/tex]3/4x-4?
Answers
Toby and Emily went to the grocery store to buy food for breakfast for the week. They needed 8 bananas, 2 loaves of bread, 1 bottle of OJ, 1 gallon of milk and a dozen eggs. Each banana was .25, each loaf of bread was 2.55, a bottle of OJ was $2.45 and dozen eggs were $3.15. How much money in total did Toby & Emily spend?
Answers
Answer:
$12.70
Step-by-step explanation:
8 x .25 = 2
2 x 2.55 = 5.10
1 x 2.45 = 2.45
1 x 3.15 = 3.15
__________
2 + 5.10 + 2.45 + 3.15 = 12.70 = $12.70
Answer:
they spent $15.85 in all.
Step-by-step explanation:
2.00 banana 5.10 bread 2.45 oj and 3.15 egg, add it all.
Brice brought two 2-liter bottles of sport drink to serve at basketball practice. How many 100-milliliter servings of sport drink did Brice bring?
A
Answers
Answer:
20 servings of 100-milliliter sports drink
Step-by-step explanation:
Hilary is making slime for a class project
He discovers that he needs 2 cups of glue for every 3 cups of water
Select all the ratios that have the same constant of proportionality as Hilary's
observation
4:6
3:6
3:12
8:12
Answers
8:12
Are the answers
The ratios that have the same constant of proportionality as Hilary's observation are 4:6 and 8:12.
Hilary's observation states that he needs 2 cups of glue for every 3 cups of water. This can be represented as the ratio 2:3.
Now, let's compare this ratio to the given options:
4:6 = 2:3
Same ratio, same constant of proportionality.
3:6 = 1:2
Different ratio, not the same constant of proportionality.
3:12 = 1:4
Different ratio, not the same constant of proportionality.
8:12 = 2:3
Same ratio, same constant of proportionality.
So, the ratios that have the same constant of proportionality as Hilary's observation are 4:6 and 8:12.
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What is the equation (y=mx+b) for the points (3,-1) and (4,7) PLZ SHOW HOW YOU GOT THE Y-INTERCEPT
Answers
Answer:
[tex]y=8x-25[/tex]
Step-by-step explanation:
The first thing to do is to find the slope. The slope is the difference in Y divided by the difference in x, or:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Here (3, -1) is going to be (x1, y1) and (4, 7) will be (x2, y2):
[tex]m=\frac{7--1}{4-3}=\frac{8}{1}=8[/tex]
The slope is 8. Now, you can plug that into the formula. To find the y-intercept, plug in any point on the line and solve for b:
[tex]y=mx+b\\-1=8(3)+b\\-1=24+b\\b=-25[/tex]
Now we have both the slope and the y-intercept so you can write the equation for the line:
[tex]y=8x-25[/tex]
You can test that either by checking y for some value of x, or by graphing.
Can someone please me help
Answers
Answer:
Step-by-step explanation:
the answer is 1.1324
One snowy day, the snow was falling at a rate of 3 inches per hour.
If it kept snowing at the same rate, now many inches would
accumulate in 24 hours
artice
соо
Answers
Answer:
72 inches
Step-by-step explanation:
You would multiply the 3 inches per hour times the 24 hours to get 72 inches of snow.
Answer:
72 Inches
Step-by-step explanation:
24x3=72
Please take a look at the picture
Answers
The points in the table lie on a line. Find the slope of the line
Answers
Answer: Slope formula: m = y2 - y1/ x2 - x1
2 - (-2)/ 1 - 0
2 + 2/1
4/1
Slope = 4
Would appreciate brainly <3
Please answer the 4 questions :)
Answers
Step-by-step explanation:
The lines or plains which contain R:
1. Parallel to SP ⇒ segment RQ2. Perpendicular to SP ⇒ segment RS3. Skew to SP ⇒ segment RK4. Plane parallel to KLM ⇒ plane RSPselect the two values of x that are roots of this equation 3x^2 + 1 = 5x
Answers
Answer:
[tex]x = \frac{5 + \sqrt{13} }{6} [/tex]
Step-by-step explanation:
[tex]3x { }^{2} + 1 = 5x[/tex]
[tex]3 {x}^{2} - 5x + 1 = 0[/tex]
Apply the quadratic formula with a=3, b=-5 and c=16.
Which of the following is the graph of the equation y = −2x2 − 3x + 4?
Answers
Answer:
From the looks of the graph, it looks like the answer will be Letter C. (The last graph).
Step-by-step explanation:
Graph the parabola using the direction, vertex, focus, and axis of symmetry.
Direction: Opens Down
Vertex: ( − 3 /4 , 41 /8 )
Focus: ( − 3 /4 , 5 )
Axis of Symmetry: x = − 3 /4
Directrix: y = 21 /4
x y
− 3 − 5
− 2 2
− 3 4
41 8
0 4
1 − 1
A committee is formed consisting of one representative from each of the 4 departments (CS-IS-IT-SW) in the computer science college, where the representative from a department is either the president or one of the two representatives from that department. How many ways are there to form this committee?
Answers
Step-by-step explanation:
each department has 3 options for their representative.
and each of these options can be associated with the 3 options of the next department. and so on.
so, we have
3×3×3×3 = 3⁴ = 81
possibilities to firm that committee overall.
A shipment contains 9 igneous, 7 sedimentary, and 7 metamorphic rocks. If 7 rocks are selected at random, find the probability that exactly 5 are sedimentary.
Answers
Answer:
75 percent
Step-by-step explanation:
see the thing is
The probability that exactly 5 are sedimentary is from a shipment which contains 9 igneous, 7 sedimentary, and 7 metamorphic rocks is 21/33649.
What is combination?Combination is the way of arrangement or the collection of items in the particular order. The order of this group of items does not matter in the combination type of arrangement.
A shipment contains 9 igneous, 7 sedimentary, and 7 metamorphic rocks. If 7 rocks are selected at random, find the probability that exactly 5 are sedimentary.
Number of ways to choose 5 sedimentary from 7 sedimentary is,
[tex]^7C_5=\dfrac{7!}{(7-5)!5!}\\^7C_5=\dfrac{7\times6\times5!}{2!5!}\\^7C_5=\dfrac{7\times6}{2\times1}\\^7C_5=21[/tex]
There are total 23 rocks (9+7+7). Number of ways to choose 7 rocks from 23 rocks is,
[tex]^{23}C_5=\dfrac{23!}{(23-5)!5!}\\^{23}C_5=\dfrac{23!}{18!5!}\\^{23}C_5=33649[/tex]
Thus, the probability that exactly 5 are sedimentary is,
[tex]P=\dfrac{21}{33649}[/tex]
Hence, the probability that exactly 5 are sedimentary is from a shipment which contains 9 igneous, 7 sedimentary, and 7 metamorphic rocks is 21/33649.
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Consider the functions f(x) and g(x). The function f(x)=2x-5 and g(x)= -f(x+1)+3. Describe the three transformations that are performed on f(x) to create g(x).
A.
1.
2.
3.
B. Sketch a graph of g(x):
Answers
Answer: 3
Step-by-step explanation:
Given the points (-2, 6) and (0, -7) find the slope.
Answers
Answer:
-13/2
or
-6.5
Step-by-step explanation:
yeah-ya.......... right?
СЕ
What is the product of
(5m2) (2m -3)
7
7m6
10
10m6
Answers
Answer:
10m^3-15m^2
Step-by-step explanation:
A motorboat is capable of traveling at a speed of 14 miles per hour in still water. On a particular day, it took 15 minutes longer to travel a distance of 12 miles
upstream than it took to travel the same distance downstream. What was the rate of current in the stream on that day?
Answers
By solving a system of equations we will find that the rate of current in the stream is S = 2 mi/h.
When the motorboat travels downstream, the total velocity will be the velocity of the motorboat in still water plus the velocity of the stream, while if the motorboat travels upstream, we have the velocity of the stream subtracted.
So upstream the speed is:
(14 mi/h - S)
Downstream the speed is:
(14 mi/h + S)
Where S is the rate of current in the stream.
We know that downstream it takes 15 minutes more to travel 12 miles, then we can write the system of equations:
(14 mi/h + S)*T = 12 mi
(14 mi/h - S)*(T - 15 min) = 12 mi
To solve this, we need to isolate one of the variables in one of the equations, I will isolate T in the first one:
T = (12 mi)/(14 mi/h + S)
Replacing that in the other equation we will get:
(14 mi/h - S)*((12 mi)/(14 mi/h + S) - 15 min) = 12 mi
Now we can solve this for S. Now we can multiply both sides by (14 mi/h + S).
(14 mi/h - S)*12 mi - (14 mi/h + S)*(14 mi/h - S)*(- 15 min) = 12 mi*(14 mi/h + S)
Also notice that the speeds are in hours, so we can rewrite:
- 15 min = -0.25 h
(14 mi/h - S)*12 mi - (14 mi/h + S)*(14 mi/h - S)*(- 0.25 h) = 12 mi*(14 mi/h + S)
168 mi^2/h - 12mi*S + 49mi^2/h + 0.25h*S^2 = 168mi^2/h + 12mi*S
- 12mi*S + 49mi^2/h - 0.25h*S^2 = 12mi*S
-24mi*S - 0.25h*S^2 + 49mi^2/h = 0
This is a quadratic equation, the solutions are:
[tex]S = \frac{24mi \pm \sqrt{(-24mi)^2 - 4*(49mi^2/h)*(-0.25h)} }{2*-0.25h} \\\\S = \frac{24mi \pm 25 mi }{-0.5h}[/tex]
We only take the positive solution, so we get:
S = (24 mi - 25 mi)/(-0.5 mi) = 2 mi/h
The rate of current in the stream is 2 mi/h.
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How have vaccines impacted health?
A: Vaccines have only made a little progress at preventing most diseases.
B: Vaccines are effective on some people but not others.
C: Large numbers of people have had allergic reactions to vaccines
D: Some diseases have almost completely disappeared
Answers
Answer:
D some disease have almost completely disappeared
Which fraction converts to a repeating decimal number?
Answers
Answer:
1/3
Step-by-step explanation:
