3. For each differential equation, find the Laplace transform of the solution y:
3y' + 5y = x^2 , y(0) = 2.
y'' − y = 5e^−4x + 2x, y(0) = y'(0) = 0.
y'' + 3y' + 2y = 5 sin(πx), y(0) = y'(0) = 0.
Answers
Answer:
See below for answer
Step-by-step explanation:
First problem
[tex]3y'+5y=x^2,\: y(0)=2\\\\3\mathcal{L}\{y'\}+5\mathcal{L}\{y\}=\mathcal{L}\{x^2\}\\\\3[sY(s)-y(0)]+5Y(s)=\frac{2}{s^{3}}\\ \\3[sY(s)-2]+5Y(s)=\frac{2}{s^{3}}\\\\3sY(s)-6+5Y(s)=\frac{2}{s^{3}}\\\\(3s+5)Y(s)-6=\frac{2}{s^3}\\ \\(3s+5)Y(s)=\frac{2}{s^3}+6\\\\Y(s)=\frac{2}{s^3(3s+5)}+\frac{6}{3s+5}\\ \\Y(s)=\frac{2+6s^3}{s^3(3s+5)}[/tex]
Perform the partial fraction decomposition
[tex]\frac{2+6s^3}{s^3(3s+5)}=\frac{A}{s}+\frac{B}{s^2}+\frac{C}{s^3}+\frac{D}{3s+5}\\ \\2+6s^3=s^2(3s+5)A+s(3s+5)B+(3s+5)C+s^3D\\\\2+6s^3=3s^3A+5s^2A+3s^2B+5sB+3sC+5C+s^3D\\\\2+6s^3=3s^3A+s^3D+5s^2A+3s^2B+5sB+3sC+5C\\\\2+6s^3=s^3(3A+D)+s^2(5A+3B)+s(5B+3C)+5C[/tex]
Solve for each constant
[tex]\begin{cases} 3 A + D = 6\\5 A + 3 B = 0\\5 B + 3 C = 0\\5 C = 2 \end{cases}[/tex]
[tex]5C=2\\C=\frac{2}{5}[/tex]
[tex]5B+3C=0\\5B+3(\frac{2}{5})=0\\5B+\frac{6}{5}=0\\5B=-\frac{6}{5}\\B=-\frac{6}{25}[/tex]
[tex]5A+3B=0\\5A+3(-\frac{6}{25})=0\\5A-\frac{18}{25}=0\\5A=\frac{18}{25}\\A=\frac{18}{125}[/tex]
[tex]3A+D=6\\3(\frac{18}{125})+D=6\\\frac{54}{125}+D=6\\D=\frac{696}{125}[/tex]
Take the inverse transform and solve for the IVP
[tex]Y(s)=\frac{\frac{18}{125}}{s}+\frac{- \frac{6}{25}}{s^{2}}+\frac{\frac{2}{5}}{s^{3}}+\frac{\frac{696}{125}}{3 s + 5}\\\\y(x)=\frac{18}{125}-\frac{6}{25}x+\frac{1}{5}x^2+\frac{232}{125}e^{-\frac{5}{3}x}[/tex]
Related Questions
Find the value of x.
Answers
Answer:
-7
Step-by-step explanation:
∠D = ∠B
2x² + 7 = x² - 8x
2x² - x² + 8x + 7 = 0
x² + 8x + 7 = 0
(x + 7)(x + 1) = 0
x = -7 or -1
For x = -7, we have
x² - 8x = 49 + 56 = 105
2x² + 7 = 2*49 + 7 = 105 Right, x = -7 is an answer
For x = -1, we have
x² - 8x = 1 + 8 = 9 < 90
In the diagram angle B is > 90, so x = -1 is not the answer.
!!!Brainliest to the first one to get it right!!!!
Which of the following is the difference of two squares?
A)8x^2-16y^
B)25m^2+100n^2
C)4g-16h
D)100a^4-9b^12
Answers
Answer: The Answer is A
Step-by-step explanation:
8x^2 - 16y^2
Answer:it’s A
Step-by-step explanation:took the test
A farmer needs to mix 12 ounces of fertilizer for every 10 pounds of soil. How many pound of fertilizer will the farmer need to make a mix that contains 120 pounds of soil?
Answers
There will be 9 pounds of fertilizer
What’s the untit rate for 400 miles on 20 gallons
Answers
Answer:
20 miles per gallon
Step-by-step explanation:
We want a unit rate where
1 is in the denominator
so we divide top and bottom by 20
[tex]\frac{400 miles/20}{20 gallons/20}[/tex] = [tex]\frac{20 miles}{1 gallon}[/tex] = 20 miles a gallon
What is the surface area of this triangle prism
Answers
Answer:
Surface area = (Perimeter of the base × Length of the prism) + (2 × Base Area) = (S1 + S2+ S3)L + bh;
Write 4 equivalent expressions to 5n = 15n
Answers
Here\/ .
PLEASE PLEASE HELP
Connor needs 60 mL of a 19% acid solution for a science experiment. He has available a 21% solution and a 15% solution.
How many milliliters of the 21% solution and how many milliliters of the 15% solution should he mix to make the 19%
solution?
Answers
Answer:
Let x = ml of 21% solution
Let y = ml of 15% solution
ATQ,
x + y = 60.....(1)
or
0.21x + 0.15y = 0.6....(2)
Multiplying equation 1st by 0.21, we get,
0.21y+0.21y =12.6....(3)
subtracting equation 3
0.21y+0.21y =12.6
-0.21x +0.15y = 0.6
_________________
ㅤㅤㅤ 0.6y = 12
_________________
6y = 12
y = 20ml
x= 60-20 = 40 ml
Hence , Conner needs 20ml of 21% solution & 40ml of 15% solution
work out the area of the parallelogram
Answers
Answer:
b x h
Step-by-step explanation:
Is negative one a rational number?
Answers
Answer: -1 is a rational number.
Step-by-step explanation: -1 is a rational number because it is an integer. An integer can be any number, such as -5, 5, 0, 3, etc.
Rational numbers can also be perfect squares (i.e square root of 25), it can be repeating decimals or decimals in general.
Irrational numbers are numbers that cannot be written as a fraction OR that are imaginary.
Irrational numbers would be repeating decimals that do not have the same values repeated, such as Pi. Pi cannot be written as a fraction, only estimated. Irrational numbers can also be imaginary numbers such as the square root of -4. There is no number in this world that can multiply itself and become -4. A negative times a negative is positive, and a positive times a positive is a positive. This means the square root -4 is imaginary and therefore irrational.
Answer:
Yes.
Step-by-step explanation:
All integers, fractions, whole numbers & natural numbers fall into the category of rational numbers.
How are integers rational numbers?
The definition of rational numbers says that any number that can be written in the form of p/q, where p & q are integers & q is not equal to 0 are categorised into rational numbers. This is in the form of the fraction. We know that, by dividing the integers by 1, we'll get them as a fraction. Sp, integers are rational numbers.
For example:
- 1 = -1/1 (p/q form; so it's a rational number)
________
Hope it helps ⚜
Quadrilateral GHIJ is similar to quadrilateral KLMN. Find the measure of side NK.
Figures are not drawn to scale.
Answers
Answer:
NK = 14.4
Step-by-step explanation:
Given the following question:
We know that the two sides are similar
JI = 3, NM = 7.2
The scale factor is 2.4 since....
[tex]3\times2.4=7.2[/tex]
Knowing the two sides are similar and if we multiply JI by 2.4, we get NM we can easily do the same for JG and NK.
JG = 6
[tex]6\times2.4=14.4[/tex]
[tex]=14.4[/tex]
NK = 14.4
Hope this helps.
joy is (4x + 5) years old her brother micheal is (3x - 2) years younger than her sister. then write an expression thet represents Michael's age
Answers
Answer:
(x +7) years old
Step-by-step explanation:
Joy= (4x +5) years old
Since Michael is (3x -2) years younger, we need to subtract (3x -2) from Joy's age to find out his age.
Michael
= (4x +5) -(3x -2)
= 4x +5 -3x +2 (expand)
= x +7
Thus, Michael is (x +7) years old.
Find the value of x, y, and z, in the rhombus below.
Answers
Answer:
Step-by-step explanation:
helppppp
Find the value of x.
Answers
The value of x is 8.77
Ivan rocks 2 1/8 miles to his aunts house he has already walked 3/4 miles how much further does he have to go explain how you solved
Answers
Someone help and please make sure the answer right pls, pls help I’m confused
Answers
Answer:
it's the third bubble
Step-by-step explanation:
Answer:
n³ - 2n² - 3n + 2 + (2 / x - 4)
Step-by-step explanation:
n^4 - 6n³ + 5n² + 14n - 6
(n - 4) x n³
n^4 - 6n³ + 5n² + 14n - 6
-n^4 + 4n³
-2n³ + 5n² + 14n - 6
(n - 4) x -2n²
-2n³ + 5n² + 14n - 6
2n³ - 8n²
-3n² + 14n - 6
(n - 4) x -3n
-3n² + 14n - 6
3n² - 12n
2n- 6
(n - 4) x 2
2n - 6
-2n + 8
R. 2 / x - 4
I hope this helps!
What is sin 60°?
A. 1
B. 1/2
Answers
Answer:
B.
[tex] \frac{1}{2} [/tex]
Hope it helps you<3Asriel says 2 times 2 is 4 Adrian says 2 times 3 is 6 Brittne says 2 times 5 is 4.Who is correct?
Answers
Answer:
Asriel is correct because 2 times 2 is 4 and 2 times 3 is 6 and 2 times 5 is 10 so Asriel is correct.
Answer: Asriel & Adrian
Step-by-step explanation: 2x2 =4 (Ex: if you have 2 apples and was given 2 more you would have 4 in total)
2x3=6 (Ex: like if you have 2 oranges and was given 3 more you always have to count by your multiples.)
PLEASE HELP!!
(2x+3y-7)+(x-y)
show how you got the answer too please
Answers
Answer:
3x +2y -7
Step-by-step explanation:
The given expression can be simplified by combining like terms. Those are terms with the same variable. The commutative and associative properties of addition apply, as well as the distributive property.
(2x +3y -7) +(x -y) . . . . . given
= 2x +3y -7 +x -y . . . . . . eliminate (unneeded) parentheses
= (2x +x) +(3y -y) -7 . . . . group like terms
= (2+1)x +(3-1)y -7 . . . . . . combine coefficients of like terms (this is an application of the distributive property)
= 3x +2y -7
PLEASE HELP IM ALMOST DONE WITH THE CLASS
Answers
Answer: B
Step-by-step explanation:
What is the area of the trapezoid? Enter your answer in the bOx
Answers
Solution:
Note that:
Area of trapezoid = Area of rectangle + 2(Area of triangle)Finding the area of the square.
Area of rectangle = LB=> Area of rectangle = (8)(6)=> Area of rectangle = 48 in²Finding the area of the 2 triangles.
Area of triangle = 1/2 x Base x Height=> Area of triangle = 1/2 x 4 x 6Since there are two triangles with the same base and height, we will multiply the area by 2 to cover the other triangle.
=> Area of 2 triangles = 2(1/2 x 4 x 6)=> Area of 2 triangles = 4 x 6=> Area of 2 triangles = 24Finding the area of the trapezoid.
Area of trapezoid = Area of rectangle + 2(Area of triangle)=> Area of trapezoid = 48 + 24=> Area of trapezoid = 72 in²what is 8 & 3/4 x 3
(I need this today!)
Answers
3/4 x 3 = 9/4
Since you need to multiply first you add the 8 later after you solve the multiplication
8 9/4
Answer:
26 [tex]\frac{1}{4}[/tex]
Step-by-step explanation:
8[tex]\frac{3}{4}[/tex] = [tex]\frac{(8x4)+3}{4}[/tex] = [tex]\frac{35}{4}[/tex]
[tex]\frac{35}{4}[/tex] x [tex]\frac{3}{1}[/tex] = [tex]\frac{105}{4}[/tex] = 26.25 = 26 [tex]\frac{1}{4}[/tex]
1. THE CAR WAS BOUGHT BY MY BROTHER. Active or passive
Answers
Answer:
The sentence is passive.
(Btw this is English lol, not mathematics)
Aysha has 136 buttons and 8containers,can she divided the buttons equally between the containers?
Answers
30 more than the quotient of b and c
Answers
Answer:
b/c+30
Step-by-step explanation:
This is what I think it is. Have a good day
PLS HELP ME OUTT!!!!
Answers
Answer:
90+36+(4x+6)=180
4x+6=54
4x=48
x=12
Jeff has four credit cards with limits of $4,000, $9,000, $3,600 and $11,300 respectively. If he has all of his cards charged up to 85% of his credit limit, how much does he owe?
Answers
Answer:
Jeff would owe $23,715
Step-by-step explanation:
$4,000+$9000+$3600+$11300x 0.85 = $23,715
There is a pair of parallel sides in the following shape.
10
2
2
What is the area of the shape?
Answers
The area of the shape is 12 square units
Area of a trapeziumThe given shape is a trapezium and the formula for calculating the area of a trapezium is expressed as:
A = 0.5(a+b)h
Given the following parameters
a = 2
b = 10
h = 2
Substituting the given parameters
A = 0.5(2+10)*2
A = 0.2(12) * 2
A = 12 square units
Hence the area of the shape is 12 square units
Learn more on area of trapezium here: https://brainly.com/question/16904048
please help me answer this asap thank you
In circle N chord CD is 18. Find the length of the radius to the nearest tenth.
Answers
The radius of the given circle is 12.1 (correct to the nearest tenth), computed using chord properties.
We assume the radius of this circle to be x.
By the theorem saying that a perpendicular from the center of the circle to a chord, bisects the chord, we can say that CE = ED = 0.5CD = 0.5*18 = 9.
Now, we know that NC = NA = radius of the circle = x.
We are given that AE = 4.
Therefore, NE = NA - AE = x - 4.
Now, in right-angled triangle NEC, by Pythagoras theorem, we can say that,
NC² = CE² + NE²
or, x² = 9² + (x - 4)²
or, x² = 81 + x² - 8x + 16
or, x² - x² + 8x = 97
or, 8x = 97
or, x = 97/8 = 12.125 ≈ 12.1 (correct to the nearest tenth, that is, rounding off up to one decimal place).
Learn more about chord properties at
https://brainly.com/question/13950364
#SPJ10
What is the trigonometric ratio for sin C?
Enter your answer, as a simplified fraction.
Answers
Answer:
(9/40)
Step-by-step explanation:
Sin is Opposite/Adjacent
sin(C) = AB/BC
We can find the value of AB by using the Pythagorean Theorem:
a^2 + b^2 = c^2
(80)^2 + b^2 = (82)^2
b = 18
sin(C) = 18/80 or (9/40)
If A = 2x - 722 + 3 and B = 4x2 - 12x, what is A + B?
A) -3x^2+10x+3
B) -3x^2-10x+3
C)3x^2-10x+3
D)6x^2-7x-9
Answers
Answer:
B
Step-by-step explanation:
Combine like terms. Like terms have same variable with same power.
2x & (-12x) are like terms and
-7x² & 4x² are like terms
A = 2x - 7x² + 3
B = 4x² - 12x
A + B = 2x - 7x² + 3 + 4x² - 12x
= 2x - 12x - 7x² + 4x² + 3
= (2-12)x + (-7 + 4)x² + 3
= -10x + (-3x²) + 3
= -3x² - 10x +3