3b. For each differential equation, find the Laplace transform of the solution y:
y'' − y = 5e^−4x + 2x, y(0) = y'(0) = 0.
Answers
Answer:
[tex]y(x)=-2x-\frac{11}{6}e^{-x}+\frac{3}{2}e^x+\frac{1}{3}e^{-4x}[/tex]
Step-by-step explanation:
[tex]y''-y=5e^{-4x}+2x,\: y(0)=y'(0)=0\\\\\mathcal{L}\{y''\}-\mathcal{L}\{y\}=\mathcal{L}\{5e^{-4x}\}+\mathcal{L}\{2x\}\\\\s^2Y(s)-sy(0)-y'(0)-Y(s)=\frac{5}{s+4}+\frac{2}{s^2}\\ \\s^2Y(s)-Y(s)=\frac{5}{s+4}+\frac{2}{s^2}\\ \\(s^2-1)Y(s)=\frac{5}{s+4}+\frac{2}{s^2}\\ \\Y(s)=\frac{5}{(s+4)(s^2-1)}+\frac{2}{s^2(s^2-1)}\\ \\Y(s)=\frac{5s^2+2(s+4)}{s^2(s+4)(s^2-1)}\\ \\Y(s)=\frac{5s^2+2s+8}{s^2(s-1)(s+1)(s+4)}[/tex]
Perform the partial fraction decomposition
[tex]\frac{5 s^{2} + 2 s + 8}{s^{2} \left(s - 1\right) \left(s + 1\right) \left(s + 4\right)}=\frac{A}{s}+\frac{B}{s^{2}}+\frac{C}{s + 1}+\frac{D}{s - 1}+\frac{E}{s + 4}\\\\5 s^{2} + 2 s + 8=s^{2} \left(s - 1\right) \left(s + 1\right) E + s^{2} \left(s - 1\right) \left(s + 4\right) C + s^{2} \left(s + 1\right) \left(s + 4\right) D + s \left(s - 1\right) \left(s + 1\right) \left(s + 4\right) A + \left(s - 1\right) \left(s + 1\right) \left(s + 4\right) B[/tex]
[tex]5 s^{2} + 2 s + 8=s^{4} A + s^{4} C + s^{4} D + s^{4} E + 4 s^{3} A + s^{3} B + 3 s^{3} C + 5 s^{3} D - s^{2} A + 4 s^{2} B - 4 s^{2} C + 4 s^{2} D - s^{2} E - 4 s A - s B - 4 B\\\\5 s^{2} + 2 s + 8=s^{4} \left(A + C + D + E\right) + s^{3} \left(4 A + B + 3 C + 5 D\right) + s^{2} \left(- A + 4 B - 4 C + 4 D - E\right) + s \left(- 4 A - B\right) - 4 B[/tex]
Solve for each constant
[tex]\begin{cases} A + C + D + E = 0\\4 A + B + 3 C + 5 D = 0\\- A + 4 B - 4 C + 4 D - E = 5\\- 4 A - B = 2\\- 4 B = 8 \end{cases}[/tex]
[tex]-4B=8\\B=-2[/tex]
[tex]-4A-B=2\\-4A-(-2)=2\\-4A+2=2\\-4A=0\\A=0[/tex]
[tex]A+C+D+E=0\\C+D+E=0\\E=-C-D[/tex]
[tex]-A+4B-4C+4D-E=5\\4(-2)-4C+4D-(-C-D)=5\\-8-4C+4D+C+D=5\\-3C+5D=13\\5D=13+3C[/tex]
[tex]4A+B+3C+5D=0\\4(0)+(-2)+3C+13+3C=0\\-2+6C+13=0\\11+6C=0\\6C=-11\\C=-\frac{11}{6}[/tex]
[tex]5D=13+3C\\5D=13+3(-\frac{11}{6})\\5D=13-\frac{33}{6}\\5D=\frac{15}{2}\\D=\frac{15}{10}\\D=\frac{3}{2}[/tex]
[tex]E=-C-D\\E=-(-\frac{11}{6})-(\frac{3}{2})\\E=\frac{11}{6}-\frac{3}{2}\\E=\frac{2}{6}\\E=\frac{1}{3}[/tex]
Take the inverse transform and solve for the IVP
[tex]Y(s)=\frac{0}{s}+\frac{-2}{s^2}+\frac{-\frac{11}{6}}{s+1}+\frac{\frac{3}{2}}{s-1}+\frac{\frac{1}{3}}{s+4}\\ \\y(x)=-2x-\frac{11}{6}e^{-x}+\frac{3}{2}e^x+\frac{1}{3}e^{-4x}[/tex]
Related Questions
can someone help me solve for b? 2(b + 3) = 12
Answers
-3 -3
2b=9
B=4.5
Answer:
[tex]b=3[/tex]
Step-by-step explanation:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation.
[tex]2*(b+3)-(12)=0[/tex]
Then solve!
Solve the differential equation (2x+y)dx+(x−2y)dy = 0
Answers
Answer:
y(x) = 1/2 (x - sqrt(5 x^2 + c_1)) or y(x) = 1/2 (x + sqrt(5 x^2 + c_1))
Step-by-step explanation:
Solve 2 x + y(x) + (dy(x))/(dx) (x - 2 y(x)) = 0:
Let P(x, y) = 2 x + y and Q(x, y) = x - 2 y.
This is an exact equation, because (dP(x, y))/(dy) = 1 = (dQ(x, y))/(dx).
Define f(x, y) such that (df(x, y))/(dx) = P(x, y) and (df(x, y))/(dy) = Q(x, y).
Then, the solution will be given by f(x, y) = c_1, where c_1 is an arbitrary constant.
Integrate (df(x, y))/(dx) with respect to x in order to find f(x, y):
f(x, y) = integral(2 x + y) dx = x^2 + x y + g(y) where g(y) is an arbitrary function of y.
Differentiate f(x, y) with respect to y in order to find g(y):
(df(x, y))/(dy) = d/(dy) (x^2 + y x + g(y)) = x + (dg(y))/(dy)
Substitute into (df(x, y))/(dy) = Q(x, y):
x + (dg(y))/(dy) = x - 2 y
Solve for (dg(y))/(dy):
(dg(y))/(dy) = -2 y
Integrate (dg(y))/(dy) with respect to y:
g(y) = integral-2 y dy = -y^2
Substitute g(y) into f(x, y):
f(x, y) = x^2 - y^2 + y x
The solution is f(x, y) = c_1:
x^2 - y^2 + y x = c_1
Solve for y:
y(x) = 1/2 (x - sqrt(5 x^2 - 4 c_1)) or y(x) = 1/2 (x + sqrt(5 x^2 - 4 c_1))
Simplify the arbitrary constants:
Answer: y(x) = 1/2 (x - sqrt(5 x^2 + c_1)) or y(x) = 1/2 (x + sqrt(5 x^2 + c_1))
John had $20. He earned $5, spent $10, earned $5 again, and then spent $3. After this series of earnings and expenses, how much money did he owe or have left?
Answers
Answer:
$17
Step-by-step explanation:
So first
The total money John have is $20
then he earned $5 which means $20+$5 = $25
After that he spent $10 = $25- $10 = $15
Again earned $5 = $15+$5=$20
At last spent $3 = $20-$3=$17
Hence the total money he owe or have left is $17
Celeste drinks at least 8 cups of water each day. She drinks her water from a bottle that holds 2 cups of water. So far, she drank 1.5 cups of water. Solve the inequality that shows the number of bottles of water Celeste should drink so that she drinks at least 8 cups of water.
Answers
Step-by-step explanation:
8 cups total a day
1 bottle holds 2 cups (Celeste drinks 4 bottles a day = 8 cups)
If she drank 1.5 bottles already, she needs to drink 2.5 bottles
Let x be the number of bottles she needs to drink to have at least 8 cups
The inequality is x >= 2.5 bottles
Find the area of the figure.
3inches
5inches
7 inches
7 inches
4 inches
Answers
Answer:
26 inches
Step-by-step explanation:
You need to add 3+5+7+7+4= 26 because you need to find the area of the figure which the inches are area to add and get how many inches in the figure. Sorry if you don't get it but the answer is 26 inches.
Solve: 80 ≥ -8(w - 8) Solve: 80 ≥ -8(w - 8) w ≥ -2 w > -2 w < -2 w ≤ -2
Answers
80 >= -8(w-8)
Use distributive property:
80 >= -8w + 64
Subtract 64 from both sides:
16 >= -8w
Divide both sides by -8:
-2 >= w
Because the division was done using a negative value the inequality gets reversed:
w >= -2
Answer : w >= -2
Evaluate ∫ sin(x) cos(x) dx by four methods.
a. The substitution u = cos(x)
b. The substitution u = sin(x)
c. The identity sin(2x) = 2 sin(x) cos(x)
d. Integration by parts.
Answers
(a) Since u = cos(x) gives du = -sin(x) dx, we have
∫ sin(x) cos(x) dx = - ∫ (-sin(x)) cos(x) dx
= - ∫ cos(x) d(cos(x))
= - ∫ u du
= - 1/2 u² + C
= -1/2 cos²(x) + C
(b) Now u = sin(x) gives du = cos(x) dx, so
∫ sin(x) cos(x) dx = ∫ sin(x) d(sin(x))
= ∫ u du
= 1/2 u² + C
= 1/2 sin²(x) + C
(c) Since sin(2x) = 2 sin(x) cos(x), we have
∫ sin(x) cos(x) dx = 1/2 ∫ sin(2x) dx
Substitute u = 2x, so that du = 2 dx, and
∫ sin(x) cos(x) dx = 1/2 ∫ sin(2x) dx
= 1/4 ∫ 2 sin(2x) dx
= 1/4 ∫ sin(u) du
= -1/4 cos(u) + C
= -1/4 cos(2x) + C
(d) Integrate by parts, setting
u = sin(x) ==> du = cos(x) dx
dv = cos(x) dx ==> v = sin(x)
Then
∫ sin(x) cos(x) dx = sin²(x) - ∫ sin(x) cos(x) dx
2 ∫ sin(x) cos(x) dx = sin²(x) + C
∫ sin(x) cos(x) dx = 1/2 sin²(x) + C
The solutions in (b) and (d) are identical, but all 4 are equivalent, and this follows from the identities,
sin²(x) + cos²(x) = 1
cos(2x) = cos²(x) - sin²(x)
Find the distance between the two points rounding to the nearest tenth (if necessary).
(-4,2) and (2,-6)
Answers
Answer:
10
Step-by-step explanation:
Distance formula:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Let [tex](x_1,y_1)[/tex] = (-4, 2)
Let [tex](x_2,y_2)[/tex] = (2, -6)
Substituting given points into the formula:
[tex]\implies d=\sqrt{(2-(-4))^2+(-6-2)^2}[/tex]
[tex]\implies d=\sqrt{(6)^2+(-8)^2}[/tex]
[tex]\implies d=\sqrt{36+64}[/tex]
[tex]\implies d=\sqrt{100}[/tex]
[tex]\implies d=10[/tex]
Divide.
1.95 -0.5
please answer asap:(
Answers
Answer:
Step-by-step explanation:
3.9
please help me who ever gets it first gets brainliest
Sofia buys 3.8 pounds of potatoes for $0.85 per pound. Flynn buys 4.2 pounds of potatoes for $0.75 per pound.
How much more does Sofia spend than Flynn?
$0.08
$0.30
$0.72
$0.80
Answers
For Sofia:-
3.8lbs=$0.85So
Unit rate=0.85/3.8=$0.224/lbsFor Flynn:-
4.2lbs=$0.75Unit rate=0.75/4.2=0.178$/lbsNow
difference
0.224-0.178=$0.46/lbsHow can you convert 12/10 into a decimal
Answers
Answer:
1.2
Step-by-step explanation:
WILL MARK AS BRAINLIEST
The points (-7,r) and (-3, 4) lie on a line with slope - 3/4.
Find the missing coordinate r
Answers
for A (xa ; ya) and B (xb ; yb)
the slope is (yb - ya) / (xb - xa)
here :
slope = - 3/4 = (4 - r) / (-3 - (-7))
(4 - r) / 4 = - 3/4
so 4 - r = - 3
then r = 7
check
points (-7 ; 7) and (-3 ; 4)
slope = (4 - 7) / (-3 - (-7) = -3/4
A 390 m wire is cut into three pieces. The second piece Is 4 times as long as the first. The third plece is 2 times as long as the second. How long is each piece?
Answers
Answer:
First piece = 30m , 2nd is 120 m and third piece is 240 m long.
Step-by-step explanation:
Let the length of the first piece be x m.
Then the length of the second piece = 4x m.
The third piece = 2 * 4x = 8x m.
So x + 4x + 8x = 390
13x = 390
x = 30 m
So 2nd piece = 4*30 and third is 8*30 m.
Evaluate
1/3^-2x^-4y^2 for x = 3 and y = -1.
Answers
Answer:
1/9
Step-by-step explanation:
First turn all the negative exponents into positive by flipping the fractions then plug in the given values.
(3/1)^2 * (1/x)^4 * y^2
Simplify to:
9 * 1/(x^4) * y^2
Plug in:
9 * 1/(3^4) * -1^2
Simplify:
9 * 1 * 1/(3^4) = 9/(3^4) = 9/81 = 1/9
Help math math math ASAP
Answers
The equation of any straight line can be written as in slope intercept form as -
[tex]\green{ \underline { \boxed{ \sf{y=mx+c}}}}[/tex]
where,
m is its slopec is its y-intercept.Given equation:-
[tex]\begin{gathered}\\\implies\quad \sf 5x+y = 10 \\\end{gathered} [/tex]
Arranging it in slope intercept form by transposing 5x to RHS
[tex]\begin{gathered}\\\implies\quad \sf y = 10 -5x \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies\quad \sf y = -5x +10\\\end{gathered} [/tex]
[tex]\leadsto[/tex]Comparing the equation y = -5x +10 with the standard form of the equation, we get -
[tex]\quad \bull\: \sf m= -5[/tex]
[tex]\quad \bull \: \sf c= 10[/tex]
Thus , 5 is our required answer.
Answer:
[tex]\displaystyle y = -5x + 10[/tex]
Step-by-step explanation:
5x + y = 10
- 5x - 5x
_________
[tex]\displaystyle \boxed{y = -5x + 10}[/tex]
I am joyous to assist you at any time.
It takes Sam 3/5 of an hour to clean his room. it takes Mia 2/3 of an hour to clean her room. Who takes longer?
Answers
Answer:
Mia takes longer
Step-by-step explanation:
We can do this easily by doing 3 / 5 which is 0.6 for Sam and for Mia 2/3 is 0.666 repeating. 0.666 repeating is greater than 0.6
What type of transformation of △ABC results in △A′B′C′ ? reflection rotation translation I don't know. Two triangles on the coordinate plane. Triangle A B C is formed by the points A at 1 comma 2, B at 3 comma negative 1, and C at 5 comma 1. Triangle A prime B prime C prime is formed by the points at A prime 2 comma negative 1, B prime at negative 1 comma negative 3, and C prime at 1 comma negative 5.
Answers
Using translation concepts, it is found that a 90º clockwise rotation of the triangle around the origin took place.
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
In this problem, considering the points on triangles ABC and A'B'C', the change is modeled by:
(x,y) -> (y, -x).
Which is a 90º clockwise rotation of the triangle around the origin.
More can be learned about translation concepts at https://brainly.com/question/4521517
.helooooooooo......... help
Answers
Answer:
B
Step-by-step explanation:
sarah and Noah work at Read
Answers
That's the question????
Lukas is able to package and ship 416 cards in an 8-hour workday. How many cards can Lukas package a ship 32 hours?
Answers
Answer:
1664 cards
Step-by-step explanation:
416 divided by 8 is 52 and 52 times 32 is 1664.
Which equation represents a line which is parallel to y = 0?
A) x = - 4y
B) x = 1/2y
C) x = 6
D) y = -5
Answers
Answer:
C.
Step-by-step explanation:
If you graph it, you can see that the lines are parallel.
Plzzzzzzzzzzzzzz help mee plzzzzzzzzzzzzzz
Answers
First, you want to find the area of the entire thing first.
The length adds up to 14, the width to 10. Multiply and you get 140 as the total area.
Now to get rid of the flower bed is simple, 8 Multiply to 3 which makes 24, then divide by 2 which is 12. Substract 12 to 140 and you get 128.
The formula for a trapezoid is Area= (base1 +base2)/2 times height. Following this, and 5 and 7 which gets 12. divide 12 to 2 and you get 6. 6 times 10 is 60.
Subtract 60 to 128 and you get 68.
Mark best if possible please.
Answer:
68Step-by-step explanation:
First, you want to find the area of the entire thing first.
The length adds up to 14, the width to 10. Multiply and you get 140 as the total area.
Now to get rid of the flower bed is simple, 8 Multiply to 3 which makes 24, then divide by 2 which is 12. Substract 12 to 140 and you get 128.
The formula for a trapezoid is Area= (base1 +base2)/2 times height. Following this, and 5 and 7 which gets 12. divide 12 to 2 and you get 6. 6 times 10 is 60.
Subtract 60 to 128 and you get 68.
.
+ + 2
- 2
A = 96
*- 2
2+2
Polynomial:
Equation:
Answers
Answer:
Length = 8
Breadth = 12
Step-by-step explanation:
Since Area = length * breadth
96 = (x+2)(x-2)
Since (a+b)(a-b) = a^2-b^2
(x+2)(x-2) = x^2-4
Therefore
x^2-4 = 96
x^2 = 100
x = +-10
Since x > 0
x = 10
A construction crew is placing a square in a wall for a new window.
The window is 3 1/8 feet wide.
The wall has a length of 11 3/8 feet.
The crew wants to position the window at the center of the wall.
How far from each edge of the wall should the window be positioned?
Answers
Answer:
4 1/3 feet
Step-by-step explanation:
11 3/8 - 3 1/8 = 8 2/3
8 2/3 / 2 = 4 1/3
How tall would a building have to be if, after hitting the ground ten times, the ball bounces to 1 m? Is there a building this tall?
Answers
The building should be 1.024 meters high and unfortunately there are no buildings with such height in the world.
How to estimate the height of a building by geometric progressionsIn this question we must use definition of geometric progression to predict initial height ([tex]h_{o}[/tex]), in meters, in terms of current height ([tex]h[/tex]), in meters, and number of bounces experimented by the ball ([tex]n[/tex]). The expression is described below:
[tex]h = h_{o}\cdot \left(\frac{1}{2} \right)^{n}[/tex] (1)
If we know that [tex]n = 10[/tex] and [tex]h = 1\,m[/tex], then the initial height is:
[tex]1 = h_{o}\cdot \left(\frac{1}{2} \right)^{10}[/tex]
[tex]h_{o} = 2^{10}[/tex]
[tex]h_{o} = 1024\,m[/tex]
The building should be 1.024 meters high and unfortunately there are no buildings with such height in the world. [tex]\blacksquare[/tex]
RemarkThe statement is incomplete, complete form is shown below:
A ball falls from the roof of a building and bounces half as high each time. How tall would a building have to be if, after hitting the ground ten times, the ball bounces to 1 meter? Is there a building this tall?
To learn more on geometric series, we kindly invite to check this verified question: https://brainly.com/question/15130111
1) State the slope and y-intercept of: y = 2x - 4*
Your answer
Answers
Answer:
slope=2 y-intercept=4
CORRECT AND SERIOUS ANSWERS ONLY ORELSE REPORT
Answers
Answer:
fraction: 98/100 = 49/50decimal: 0.98percent: 98%Step-by-step explanation:
The fraction will represent the number of shaded squares over the total number of squares. You have correctly identified it as 98/100. That fraction can be reduced, if you like, to 49/50.
The pronunciation of your fraction is "ninety-eight hundredths." The decimal number with that same pronunciation is 0.98.
The symbol % and the symbol /100 have identical meaning. This makes them essentially interchangeable.
98/100 = 98%
Your numbers are ...
98/100 (or 49/50), 0.98, and 98%.
_____
Additional comment
We have shown you that % is a shorthand way to write /100. There is another similar symbol: ‰ is shorthand for /1000.
1/100 is one percent. 1/1000 is one mil.
Monica is mixing some custom-colored paint for her house. The color she wants requires 3∕4 ounce of blue dye and 2∕3 of an ounce of pink dye mixed with each gallon of white paint base. How much dye is needed to mix 8 gallons of paint?
Answers
Determine the type of transformation.
Two shapes that look like the letter L. The first shape is the letter L with black continuous lines and the L points to the East. The second shape is also the letter L and it is red with dotted lines, and it is attached to the top of the first L but this L is pointing upwards. Two shapes that look like the letter L. The first shape is the letter L with black continuous lines and the L points to the East. The second shape is also the letter L and it is red with dotted lines, and it is attached to the top of the first L but this L is pointing upwards.
reflection
rotation
translation
none of these
Answers
pls help me
Use the tools below to construct a triangle with angle measures of 15°, 15°, and 150°, if possible. If there is no such triangle possible, click on "No triangle possible". Note: It is okay to have extended sides on your triangle. No triangle possible
Answers
Step-by-step explanation:
the answer is in the picture