I'm going to assume that "−" is supposed to be some kind of minus character, so that the given system of DEs is supposed to be
[tex]\begin{cases}\dfrac{\mathrm d^2x}{\mathrm dt^2} + x - y = 0 \\ \dfrac{\mathrm d^2y}{\mathrm dt^2} + y - x = 0 \\ x(0) = 0, x'(0) = -6, y(0) = 0, y'(0) = 1\end{cases}[/tex]
Take the Laplace transform of both sides of both equations. Recall the transform for a second-order derivative,
[tex]L_s\left\{\dfrac{\mathrm d^2f(t)}{\mathrm dt^2}\right\} = s^2 F(s) - sf(0) - f'(0)[/tex]
where F(s) denotes the transform of f(t). You end up with
[tex]\begin{cases}s^2X(s)-sx(0)-x'(0) + X(s) - Y(s) = 0 \\ s^2Y(s) - sy(0) - y'(0) + Y(s) - X(s) = 0\end{cases} \\\\ \begin{cases}(s^2+1)X(s) - Y(s) = -6 \\ (s^2+1)Y(s) - X(s) = 1\end{cases}[/tex]
and solving for X(s) and Y(s) (nothing tricky here, just two linear equations) gives
[tex]X(s) = -\dfrac{6s^2+5}{s^2(s^2+2)} \text{ and } Y(s) = \dfrac{s^2-5}{s^2(s^2+2)}[/tex]
Now solve for x(t) and y(t) by computing the inverse transforms. To start, split up both X(s) and Y(s) into partial fractions.
• Solving for x(t) :
[tex]-\dfrac{6s^2+5}{s^2(s^2+2)} = \dfrac as + \dfrac b{s^2} + \dfrac{cs+d}{s^2+2} \\\\ -6s^2-5 = as(s^2+2) + b(s^2+2) + (cs+d)s^2 \\\\ -6s^2-5 = (a+c)s^3 + (b+d)s^2 + 2as + 2b \\\\ \implies \begin{cases}a+c=0\\b+d=-6\\2a=0\\2b=-5\end{cases} \implies a=c=0, b=-\dfrac52, d=-\dfrac72[/tex]
[tex]\implies X(s) = -\dfrac5{2s^2} - \dfrac7{2(s^2+2)}[/tex]
Taking the inverse transform, you get
[tex]L^{-1}_t\left\{X(s)\right\} = -\dfrac52 L^{-1}_t\left\{\dfrac{1!}{s^{1+1}}\right\} - \dfrac7{2\sqrt2} L^{-1}_t\left\{\dfrac{\sqrt2}{s^2+(\sqrt2)^2}\right\} \\\\ \implies \boxed{x(t) = -\dfrac52 t - \dfrac7{2\sqrt2}\sin(\sqrt2\,t)}[/tex]
• Solving for y(t) :
[tex]\dfrac{s^2-5}{s^2(s^2+2)} = \dfrac as + \dfrac b{s^2} + \dfrac{cs+d}{s^2+2} \\\\ s^2 - 5 = as(s^2+2) + b(s^2+2) + (cs+d)s^2 \\\\ s^2 - 5 = (a+c)s^3 + (b+d)s^2 + 2as + 2b \\\\ \implies \begin{cases}a+c=0\\b+d=1\\2a=0\\2b=-5\end{cases} \implies a=c=0, b=-\dfrac52, d=\dfrac72[/tex]
[tex]\implies Y(s) = -\dfrac5{2s^2} + \dfrac7{2(s^2+2)}[/tex]
Inverse transform:
[tex]L^{-1}_t\left\{Y(s)\right\} = -\dfrac52 L^{-1}_t\left\{\dfrac{1!}{s^{1+1}}\right\} + \dfrac7{2\sqrt2} L^{-1}_t\left\{\dfrac{\sqrt 2}{s^2+(\sqrt2)^2}\right\} \\\\ \implies \boxed{y(t) = -\dfrac52 t + \dfrac7{\sqrt2} \sin(\sqrt2\,t)}[/tex]
What inequality symbol belongs in the circle: -4 _ 5
30 POINTS!
Answer:
<
Step-by-step explanation:
Inequality symbols:
< less than
> greater than
= equal to
≤ less than or equal to
≥ greater than or equal to
In this case, it is known that -4 is less than 5, or:
-4 < 5.
~
Answer:
>
Step-by-step explanation:
because 5 is bigger than -4
find the solutions to the equation
-3y(2y+5)=0
Answer:
The solutions are y=0, -5/2
Step-by-step explanation:
-3y(2y+5)=0
We can use the zero product property
-3y =0 2y+5 =0
y =0 2y = -5
y = -5/2
The solutions are y=0, -5/2
What is the total cost of a $28 pair of jeans if the sales tax is 7.5%?
Answer:
30.10
Step-by-step explanation:
First find the amount of tax
28 * 7.5%
28 * .075
2.10
Add this to the price of the pants
28+2.10 =30.10
Divide (3x^4-2x^3+4x-5) / (x^2+4)
3x ⁴ = 3x ² • x ². Then
(3x ⁴ - 2x ³ + 4x - 5) - 3x ² (x ² + 4) = -2x ³ - 12x ² + 4x - 5
-2x ³ = -2x • x ². Then
(-2x ³ - 12x ² + 4x - 5) - (-2x) (x ² + 4) = -12x ² + 12x - 5
-12x ² = -12 • x ². Then
(-12x ² + 12x - 5) - (-12) (x ² + 4) = 12x + 43
So we've shown
[tex]\displaystyle \frac{3x^4-2x^3+4x-5}{x^2+4} = 3x^2 - \frac{2x^3+12x^2-4x+5}{x^2+4} \\\\ = 3x^2 - 2x - \frac{12x^2-12x+5}{x^2+4} \\\\ = \boxed{3x^2 - 2x - 7 + \frac{12x+43}{x^2+4}}[/tex]
Listed below are the amounts of weight change (in pounds) for 12 women during their firstyear of work after graduating from college. Positive values correspond to women whogained weight and negative values correspond to women who lost weight. What is themean weight change
Answer:
Divide. Write the quotient in lowest terms.
4\dfrac{4}{5} \div 1\dfrac12 =4
5
4
÷1
2
Step-by-step explanation:
Which is the equation of the line with slope o passing through the point (-3, 6)?
Answer:
y = 6
Step-by-step explanation:
I'm assuming you are trying to say slope "0".
If the slope is 0, it means that the general equation Y = mX + c equals to Y = 0x + c and finally Y = c. Since the line is horizontal because it is 0 slope, Y = 6
The equation of the line with slope 0 passing through the point (-3, 6) will be y = 6.
What is a line segment?
A line section that can connect two places is referred to as a segment.
In other words, a line segment is just part of a big line that is straight and going unlimited in both directions.
The equation of the line passing through (x₁, y₁ ) is given by
y - y₁ = m(x - x₁)
The line is here! It extends endlessly in both directions and has no beginning or conclusion.
Given that
The line has points ( -3, 6 )
Slope(m) = 0
The equation of the line passing through (x₁, y₁ ) is given by
y - y₁ = m(x - x₁)
so,
y - 6 = 0 (x + 3)
y - 6 = 0
y = 6
Hence, the line which is passing through the point ( -3, 6 ) will be y = 6.
For more about line segment
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Simplify: - 6y^2 + 8y^3 - 4y^2 + y - 10y^3
Answer:
[tex]-6y^{2}+8y^{3}-4y^{2}+y-10y^{3}\\\\=(-6-4)y^{2}+(8-10)y^{3}+y\\\\=-2y^{3}-10y^{2}+y\\\\=y(-2y^{2}-10y+1)[/tex]
a number t is less than 7
Answer:
6 or 5 or 4...........
Step-by-step explanation:
Answer:
x=N :x<7
Step-by-step explanation:
123456
hope it's helpful
if a circumference of a circle is 22cm.find it diameter take pie 22/7.
Answer:
➕
Step-by-step explanation:
i know the answer ok it is easy
(5.5 * 10^-6 + 6.3 * 10^-6)^2
9514 1404 393
Answer:
exact: 1.3924×10^-10rounded to 2 sf: 1.4×10^-10Step-by-step explanation:
The factor 10^-6 can be brought out:
((5.5 +6.3)×10^-6)^2 = (11.8^2)×10^-12 = 1.3924×10^-10
Often, a problem like this wants you to round to a precision appropriate to the given values. Here, that is 2 significant figures:
≈ 1.4×10^-10
_____
Additional comment
Your scientific or graphing calculator can do this nicely, even showing you the result in scientific notation. A spreadsheet can do the same.
The usual rules of exponents apply.
(a^b)(a^c) = a^(b+c)
(ab)^c = (a^c)(b^c)
(a^b)^c = a^(bc)
△DOG ~△?
Complete the similarity statement and select the theorem that justifies your answer.
**If they are not similar, select "none" for both parts
9514 1404 393
Answer:
nonenoneStep-by-step explanation:
The reduced side ratios, shortest to longest are ...
AC : AT : CT = 8 : 9 : 15
OD : OG : DG = 5 : 6 : 10
These are different ratios, so the triangles are not similar.
five hundred seven billion,six hundred forty million,seven hundred forty-two thousand,seventy two
Answer:
507,640,142,072
Step-by-step explanation:
not sure what you are asking but I hope this helps! :D
Choose the correct simplification of
fh23
fh17
Answer:
C
Step-by-step explanation:
Answer:
f^6h^6
Step-by-step explanation:
f^9 h^23
-----------------
f^3 h^17
We know a^b / a^c = a^(b-c)
f^9 / f^3 = f^(9-3) = f^6
h^23 / h^17 = h^(23-17) = h^6
f^6h^6
.angle bisector of any angle of a triangle bisect that angle is_____________ parts
By definition of bisector, the angle bisector of any angle of a triangle divides that angle in two equal parts.
But first you must know the definition of the bisector of a triangle. The bisector of a triangle is a segment that divides one of its interior angles into two equal parts and continues until it reaches the side opposite that angle.
In other words, the bisector of a triangle is a line that divides each interior angle of the triangle into two equal angles.
Each interior angle of the triangle corresponds to a bisector, so since each triangle has three interior angles, it has three bisectors. The three bisectors of a triangle meet at a point called the incenter and it is always an interior point of the triangle.
In summary, the bisector of any angle of a triangle is a segment that divides the angle into two equal parts.
Learn more about angle bisector of a angle of a triangle:
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(x+y)^2 where x= 4 and y= -3
Calculate the total number of trials for an absolute frequency of five trials and a relative
frequency of 50 percent
Answer:
23456789990000000000000000000
a factory produce 1188440 bulbs in one year (365 days) how many bulbs did it produce in the month of August
BULBS PRODUCED PER DAY:-
[tex]\\ \sf\longmapsto \dfrac{1188440}{365}[/tex]
[tex]\\ \sf\longmapsto 3256[/tex]
Total days in August=31Total bulbs
[tex]\\ \sf\longmapsto 31(3256)[/tex]
[tex]\\ \sf\longmapsto 100936[/tex]
What is the smallest number which if divided by 12, 15, 20 or 54 leaves in each case a remainder 4?
Answer:
544
.....................
Answer:
543
Step-by-step explanation:
The required number = 540 + 3 = 543
hope this helps
What is 2x-40y=73 and 7x+65y=332?
[tex]\large\mathfrak{{\pmb{\underline{\orange{Given }}{\orange{:}}}}}[/tex]
[tex]2x - 40y = 73...(i)[/tex]
[tex]7x + 65y = 332...(ii)[/tex]
[tex]\large\mathfrak{{\pmb{\underline{\pink{To\:find }}{\pink{:}}}}}[/tex]
The values of [tex]x[/tex] and [tex]y[/tex].
[tex]\large\mathfrak{{\pmb{\underline{\green{Solution }}{\green{:}}}}}[/tex]
[tex]x=43.96 [/tex] and [tex]y = 0.373 [/tex].
[tex]\large\mathfrak{{\pmb{\underline{\purple{Step-by-step\:explanation}}{\purple{:}}}}}[/tex]
Let us solve this by substitution method.
From [tex]eqn.\:(i),\:we\:have [/tex]
↬[tex]2x - 40y = 73[/tex]
↬[tex]2x = 73 + 40y[/tex]
↬[tex]x = \frac{73 + 40y}{2}...(iii) \\ [/tex]
Substituting the value of [tex]x[/tex] in [tex]eqn.\:(ii)[/tex] gives us
↬[tex]7( \frac{73 + 40y}{2} ) + 65y = 332 \\ [/tex]
↬[tex] \frac{511 + 280y}{2} + \frac{65y \times 2}{1 \times 2} = 332 \\ [/tex]
↬[tex] \frac{511 + 280y + 130y}{2} = 332 \\ [/tex]
↬[tex]410y + 511 = 332 \times 2[/tex]
↬[tex]410y = 664 - 511[/tex]
↬[tex]y = \frac{153}{410} \\ [/tex]
↬[tex]y = 0.373[/tex]
Now, plug the value of [tex]y[/tex] in [tex]eqn.\:(i)[/tex]
↬[tex]2x - 40 \times 0.373 = 73 \\ [/tex]
↬[tex]2x - 14.92= 73 [/tex]
↬[tex]2x = 73 +14.92[/tex]
↬[tex]x = \frac{87.92}{2} \\ [/tex]
↬[tex]x = 43 .96[/tex]
Therefore, the values of [tex]x[/tex] and [tex]y[/tex] are [tex]\boxed{ 43.96 }[/tex] and [tex]\boxed{ 0.373 }[/tex] respectively.
[tex]\red{\large\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese \: \: Mystique35}}}}}[/tex]
a collection of tasty fruit
Answer:
Yummy!
Step-by-step explanation:
Solve for x
(x-5) ²+5 =49
Answer:
x = 5±2 sqrt(11)
Step-by-step explanation:
(x-5) ²+5 =49
Subtract 5 from each side
(x-5) ²+5-5 =49-5
(x-5) ² =44
Take the square root of each side
sqrt((x-5) ²) =±sqrt(44)
sqrt((x-5) ²) =±sqrt(4*11)
x-5 = ±2 sqrt(11)
Add 5 to each side
x-5+5 = 5±2 sqrt(11)
x = 5±2 sqrt(11)
Answer:
[tex]\left(x-5\right)^2+5=49[/tex]
Subtract 5 from both sides
[tex]\left(x-5\right)^2+5-5=49-5[/tex]
[tex]\left(x-5\right)^2=44[/tex]
[tex]x-5=\sqrt{44}[/tex]
[tex]44=2^{2} \times11[/tex]
[tex]x-5=\sqrt{11} \sqrt{2^{2} }[/tex]
Radical rule:- [tex]\sqrt[n]{a^{n} } =a[/tex]
[tex]x-5=2\sqrt{11}[/tex]
Add 5 to both sides
[tex]x=2\sqrt{11}+5[/tex]
-------
[tex]x-5=-2\sqrt{11}[/tex]
Add 5 to both sides
[tex]x-5=-2\sqrt{11}[/tex]
Ans: [tex]x=2\sqrt{11}+5,\:x=-2\sqrt{11}+5[/tex]
OAmalOHopeO
Please help me answer this?
9514 1404 393
Answer:
h'(3) = 26
Step-by-step explanation:
h(x) = 3f(x) +5g(x) +3 . . . . . . . given
h'(x) = 3f'(x) +5g'(x) . . . . . . . . take the derivative
h'(3) = 3f'(3) +5g'(3) . . . . . . . use x=3
h'(3) = 3(2) +5(4) = 26
5 pencils for 15 dollers how much for each pencil
5 pencils = 15 dollors
1 pencil = 15/5 dollors
therefore 1 pencil = 3 dollors...
Answer:
Each pencil would cost $3.00
Step-by-step explanation:
If 5 costs $15, to find the cost of one you divide 15 by 5
15 ÷ 5 = 3
The probability of drawing a red candy at random from a bag of 25 candies is 2/5.
After 5 red candies are removed from the bag, what is the probability of randomly drawing a red candy from the bag?
Please include an explanation!
Answer:
1/4
Step-by-step explanation:
Probability of red * number of candies
2/5 * 25 = 10
There are 10 red candies
25-10 = 15
There are 15 other candies
Remove 5 red
10-5 = 5 red candies
Now we have 5 red candies and 15 other candies= 20 candies
P(red) = red/total = 5/20 = 1/4
Please answer this and show the work/explain for me
2/7m - 1/7 = 3/14
this is confusing ok so 1.if there r 2 boys in a class for every 3 girls what would be the ratio for it and 2.if Seth bought a 12-ounce jar of something that is $3.60 what is the unit price?
Addition prop of equality
subtraction prop of quality
multiplication prop of equality
Division prop of equality
simplifying
distrib prop
15% of 80 is 60% of what number? There were no answer choices please help!
Answer:
80
Step-by-step explanation:
15 percent of 80 is 12
and 12 is 60 percent of 80!
hope this helps :)
Determine whether (a) x = -1 or (b) x = 2 is a solution to this equation
Answer:
2x-1=3
2x=3+1
2x=4
x=2
[tex]\huge\boxed{\underline{\bf \: Answer}}[/tex]
(a) Let's try with x = - 1
[tex] \sf \: 2x - 1 = 3 \\\sf 2( - 1) - 1 = 3 \\ \sf- 2 - 1 = 3 \\ \\ \boxed{\bf- 3 \: \bcancel= \: 3}[/tex]
So, x = - 1 is not the solution to the given equation.
______________
(b) Now, try with x = 2
[tex]\sf2x - 1 = 3 \\ \sf2(2) - 1 = 3 \\ \sf4 - 1 = 3 \\ \\ \boxed{\bf3 = 3}[/tex]
Yes, we can see that x = 2 is the correct solution for the equation.
______________
Hope it helps.
RainbowSalt2222
Find all complex solutions of 2x^2+x+6=0. (If there is more than one solution, separate them with commas.)
Answer:
x=-1/4+i*sqrt(47)/4 and x=x=-1/4-i*sqrt(47)/4
Step-by-step explanation:
Using quadratic formula, x=(-1±sqrt(1-48))/4.
x=-1/4+i*sqrt(47)/4 and x=x=-1/4-i*sqrt(47)/4
Answer:
If you do not understand any steps, please feel free to comment down below.
