We're given the ODE,
y'' + 4y' + 3y = 8t exp(-t ) + 6 exp(-t ) - (9t + 6)
(where I denote exp(x) = eˣ )
First determine the characteristic solution:
y'' + 4y' + 3y = 0
has characteristic equation
r ² + 4r + 3 = (r + 1) (r + 3) = 0
with roots at r = -1 and r = -3, so the characteristic solution is
y = C₁ exp(-t ) + C₂ exp(-3t )
For the non-homogeneous equation, assume two ansatz solutions
y₁ = (at ² + bt + c) exp(-t )
and
y₂ = at + b
• y'' + 4y' + 3y = 8t exp(-t ) + 6 exp(-t ) … … … [1]
Compute the derivatives of y₁ :
y₁ = (at ² + bt + c) exp(-t )
y₁' = (2at + b) exp(-t ) - (at ² + bt + c) exp(-t )
… = (-at ² + (2a - b) t + b - c) exp(-t )
y₁'' = (-2at + 2a - b) exp(-t ) - (-at ² + (2a - b) t + b - c) exp(-t )
… = (at ² + (b - 4a) t + 2a - 2b + c) exp(-t )
Substitute them into the ODE [1] to get
→ [(at ² + (b - 4a) t + 2a - 2b + c) + 4 (-at ² + (2a - b) t + b - c) + 3 (at ² + bt + c)] exp(-t ) = 8t exp(-t ) + 6 exp(-t )
(at ² + (b - 4a) t + 2a - 2b + c) + 4 (-at ² + (2a - b) t + b - c) + 3 (at ² + bt + c) = 8t + 6
4at + 2a + 2b = 8t + 6
→ 4a = 8 and 2a + 2b = 6
→ a = 2 and b = 1
→ y₁ = (2t ² + t ) exp(-t )
(Note that we don't find out anything about c, but that's okay since it would have gotten absorbed into the first characteristic solution exp(-t ) anyway.)
• y'' + 4y' + 3y = -(9t + 6) … … … [2]
Compute the derivatives of y₂ :
y₂ = at + b
y₂' = a
y₂'' = 0
Substitute these into [2] :
4a + 3 (at + b) = -9t - 6
3at + 4a + 3b = -9t - 6
→ 3a = -9 and 4a + 3b = -6
→ a = -3 and b = 2
→ y₂ = -3t + 2
Then the general solution to the original ODE is
y(t) = C₁ exp(-t ) + C₂ exp(-3t ) + (2t ² + t ) exp(-t ) - 3t + 2
Use the initial conditions y (0) = 2 and y' (0) = 2 to solve for C₁ and C₂ :
y (0) = C₁ + C₂ + 2 = 2
→ C₁ + C₂ = 0 … … … [3]
y'(t) = -C₁ exp(-t ) - 3C₂ exp(-3t ) + (-2t ² + 3t + 1) exp(-t ) - 3
y' (0) = -C₁ - 3C₂ + 1 - 3 = 2
→ C₁ + 3C₂ = -4 … … … [4]
Solve equations [3] and [4] to get C₁ = 2 and C₂ = -2. Then the particular solution to the initial value problem is
y(t) = -2 exp(-3t ) + (2t ² + t + 2) exp(-t ) - 3t + 2
8. Find the GCF of *
40x3 and 50x2
Answer:
2
Step-by-step explanation:
Answer:
2
Step-by-step explanation:
i will give brainliest<3
Answer:
option A = 40(24) = option B = x (.1) where x is her gross sales
40(24) = x(.1)
40(24)/.1 = x = sales = $9600
Hope this helps....
Answer:1,920
Step-by-step explanation: First find the 24$ wage. which would be 960$. then you divide that by .5 which would equal
1920
Find the volume of a cylinder whose radius is 4 and it’s height is 8
Answer:
[tex]128\pi[/tex]
Step-by-step explanation:
Formula of cylinder: [tex]\pi r^{2} h[/tex]
[tex](\pi )(4^{2}) (8)[/tex]=128pi
n a science experiment, the initial temperature was 55 degrees Fahrenheit, and each hour after, the temperature dropped by 4 degrees. Which expression represents the temperature, in degrees Fahrenheit, after t hours?
Answer:
f(t) = 55 = 4t
Step-by-step explanation:
plz somebody help me i’ll give you 40 points
Answer:
-3+-2
Step-by-step explanation:
Lines l and M are parallel. Line l can be represented by the equation 3x+4y=8. If line m passes through the point (2, -5), which equation can be used to find the equation of line m?
Answer:
3y - 4x + 7 = 0
Step-by-step explanation
The equation of the line l in point slope form is expressed as;
y-y0 = m(x-x0)
Get the slope m of the known line
Given the line 3x+4y=8
4y = -3x+8
y = -3x/4 + 2
m = -3/4
The slope of the line m = -1/(-3/4) = 4/3
Substitute m = 4/3 and the point (2,-5) into the expression above;
y - (-5) = 4/3(x-2)
y+5 = 4/3 x + 8/3
Multiply through by 3;
3y + 15 = 4x + 8
3y-4x + 15 - 8 = 0
3y - 4x + 7 = 0
Hence the equation that can be used to find the equation of line m is 3y - 4x + 7 = 0
Equation that can be used to find the equation of line m is
[tex]y+5=\frac{-3}{4} (x-2)[/tex]
Given :
Lines l and M are parallel. Line l can be represented by the equation 3x+4y=8
Line m passes through the point (2, -5)
Equation of line L is [tex]3x+4y=8[/tex]
Lets find out the slope of the given equation by writing it in y=mx+b form
where m is the slope
[tex]3x+4y=8\\4y=-3x+8\\y=\frac{-3}{4}x +2[/tex]
the slope of line L is [tex]\frac{-3}{4}[/tex]
Slope of parallel lines are equal
slope of parallel line M is also [tex]\frac{-3}{4}[/tex]
Line M passes through (2,-5)
We use point slope formula to find equation of line m
[tex]y-y_1=m(x-x_1)\\y+5=\frac{-3}{4} (x-2)\\[/tex]
Multiply both sides by 4
[tex]4(y+5)=-3 (x-2)\\4y+20=-3x+6\\3x+4y+20=6\\3x+4y=14[/tex]
equation can be used to find the equation of line is
[tex]y+5=\frac{-3}{4} (x-2)[/tex]
Learn more : brainly.com/question/19729250
A rectangular prism is made up of 36 unit cubes and is 3 unit cubes high choose two possible combinations of width and length for the prism
A) 2 cubes wide and 6 cubes long
B) 3 cubes wide and 12 cubes long
C) 4 cubes wide and 3 cubes long
D) 5 cubes wide and 7 cubes long
E) 6 cubes wide and 6 cubes long
I think (i said think) the answer is E
Which figure shows a line of symmetry?
Answer:
C)
Step-by-step explanation:
The other ones do have symmetry but it's not in the same direction
(not entirely sure but the way they are shown this makes the most sense)
Hope this helped! :)
Find the exterior angle measure:
47°
74°
Answer:
x = 121°Step-by-step explanation:
The exterior angle is equal to the sum of non-adjacent interior angles
x = 74° + 47°x = 121°100+10-8+60-80+700+20-150
Final Answer: 652
Steps/Explanations/Reasons:
Question: 100 + 10 - 8 + 60 - 80 + 700 + 20 - 150
Step 1: Simplify [tex]100 + 10[/tex] to [tex]110[/tex].
[tex]110 - 8 + 60 - 80 700 + 20 - 150[/tex]
Step 2: Simplify [tex]110 - 8[/tex] to [tex]102[/tex].
[tex]102 + 60 - 80 + 700 + 20 - 150[/tex]
Step 3: Simplify [tex]102 + 60[/tex] to [tex]162[/tex].
[tex]162 - 80 + 700 + 20 - 150[/tex]
Step 4: Simplify [tex]162 - 80[/tex] to [tex]82[/tex].
[tex]82 + 700 + 20 - 150[/tex]
Step 5: Simplify [tex]82 + 700[/tex] to [tex]782[/tex].
[tex]782 + 20 - 150[/tex]
Step 6: Simplify [tex]782 + 20[/tex] to [tex]802[/tex].
[tex]802 - 150[/tex]
Step 7: Simplify.
[tex]652[/tex]
~I hope I helped you :)~
Find the roots of x^2 + 12x + 36 + 25 = 0.
PLEASE HELP FAST ILL FIVE YOU ALL THE REST OF MY POINTS FOR THIS ONE ANSWER CHEKC IT FIRST BEFORE YOU ANSWER!!!
Answer:
They intersect at (-4,-3)
Step-by-step explanation:
hope that helps
What percent of 88 is 154?
Will mark brainliest for the first correct answer!
Answer:
the answer is 175%
Step-by-step explanation:
I hope this helped and if it did I would appreciate it if you marked me Brainliest. Thank you and have a nice day!
I got the answer wrong, this apparently is the correct answer. My deepest apologies.
57.14% is the answer that I put and it was incorrect.
PLEASE HELP!!!! ITS DUE TODAY!!! WILL GIVE BRAINLIEST!!
Answer:
ITS B
Step-by-step explabnation:
KEY WORD "PER" SO 20 ASSOCIATES WITH MULTIPLICATION
7/12 x 1/8 Find the product of in simplified form
Answer:
7/96
Step-by-step explanation:
to find the answer you need to multiply both numerators together and put that over the product of the denominators, this is going to leave you with a fraction smaller than the original fractions. that is normal, because you are multiplying terms smaller than one together. same applies to decimals less than one
Hey!! Can you also help me with this thank you−3(2y − 9) − (5y + 4)
Answer:
Let's simplify step-by-step.
3(2y−9)−(5y+4)
Distribute the Negative Sign:
=3(2y−9)+−1(5y+4)
=3(2y−9)+−1(5y)+(−1)(4)
=3(2y−9)+−5y+−4
Distribute:
=(3)(2y)+(3)(−9)+−5y+−4
=6y+−27+−5y+−4
Combine Like Terms:
=6y+−27+−5y+−4
=(6y+−5y)+(−27+−4)
=y+−31
Answer:
=y−31
Step-by-step explanation:
Joe solved the equation 3 + x/2 = 10 and justified each step as shown. Identify Joe's error and fix his mistake. ( Only answer if you truly know how. No Plagiarism and please don't reposted another students answer on brianly. Will Mark Brainliest).
Happy Holidays! :)
He swapped the Subtraction Property of Equality and the Multiplication Property of Equality.
The proper steps in order would be as shown:
3 + x / 2 = 10 Given equation
x / 2 = 7 Subtraction Property of Equality (-3 from both sides)
x = 14 Multiplication Property of Equality (x2 on both sides)
what is the value of x?
Answer:
x=4
Step-by-step explanation:
By SAS we know that
[tex]\triangle ABE \sim \triangle ADC[/tex]
by a ratio of
[tex]1:2[/tex]
so we have
[tex]\frac{BE}{DC}=\frac{1}{2}\\\\2BE=DC\\\\2(x+2)=12\\\\2x+4=12\\\\2x=12-4\\\\2x=8\\\\x=4[/tex]
Standard tests, such as the SAT or ACT or MCAT, tend to make extensive use of multiple-choice questions because they are easy to grade using software. If one such multiple choice question has possible correct answers of a, b, c, d, e, what is the probability of a wrong answer if the answer is a random guess
Answer:
4/5
Step-by-step explanation:
Given that:
Number of options = a, b, c, d, e
Number of correct answer choice = 1
Therefore, number of wrong answers = 4
Recall :
Probability = (number required outcome / number of total possible outcomes)
P(wrong answer) :
Number of Required outcome = number of wrong answer choices = 4
Total answers = 5
Hence,
P(wrong answer) = 4 / 5 = 0.8
A veterinarian knows that the supplement glucosamine is beneficial for dogs’ joints. He wants to determine if a supplement with both glucosamine and chondroitin would be more beneficial than glucosamine alone for dogs’ joint health. The veterinarian asks the owners of dogs with known joint issues if they would be willing to have their dog participate in his study, and 32 owners agree to the study. Sixteen dogs are randomly assigned to the glucosamine group and the remaining 16 are assigned to the glucosamine with chondroitin group. After 6 months, the dogs’ joint health is measured for each treatment and compared. The results of the study were inconclusive and the veterinarian decides to do the experiment again. He gets 24 volunteers and decides that he needs to use a randomized block design.
Which variable would not be an appropriate variable for blocking?
age
size
color of coat
breed
Answer:
Colour of coat
Step-by-step explanation:
We are told that the veterinarian wants to determine if a supplement with both glucosamine and chondroitin would be more beneficial than glucosamine alone for dogs’ joint health.
In randomized block design, blocking is the arrangement of experimental units in groups known as blocks which are similar to one another.
Looking at the options given, by blocking on size, age or dog breed, the source of variability which led to the inconclusiveness initially would be controlled which then leads to a greater accuracy in result. Since there can be big, medium and small size of dog: there can be older or younger dogs: there can be different type of dog breeds as well. All these can likely affect the results of the experiment.
Whereas, blocking on colour of coat is not an appropriate variable because it has no effect on the experiment.
reduce 384 by 25%
i neeeed help
Answer:
288
Step-by-step explanation:
384 *.25=96
384-96=288
The sum of three consecutive even numbers is 228. What is the 3rd number?
Answer:
78
Step-by-step explanation:
x+(x+2)+(x+4)=228
x+x+2+x+4=228
3x+6=228
Subtract 6 from both sides
3x=228-6
3x=222
Divide both sides by 3
x=74.The first number is 74
The second number (x+2)=74+2=76
The third number (x+4)=74+4=78
complete all 3 questions.
explain how you got your answer please.
if you dont explain i will not mark BRAINLIEST
giving 50 points
Answer:
1.x^2-5x
2.2x^2+15x+18
3. x^2+20x+18
Step-by-step explanation:
1. x(x-5) x*x-5*x = x^2-5x
2. (x+6)(2x+3) x*2x+x*3+6*2x+6*3= 2x^2+15x+18
3.(2x^2+15x+18)-(x^2-5x)=x^2+20x+18
Add 4/7+2/7. What is the sum in the lowest terms?
Answer:
6/7
Step-by-step explanation:
you already have a common denominator, so 4/7 plus 2/7 is 6/7
Answer:
[tex]\boxed{\frac{6}{7} }[/tex]
Step-by-step explanation:
[tex]\frac{4}{7} + \frac{2}{7}[/tex]
[tex]= \frac{4 + 2}{7}[/tex]
[tex]= \frac{6}{7}[/tex]
Write and solve an equation to find x. Then find the measure of angle YXZ.
x=
The measure of Angle YXZ is
please i have 2 minutes left!!!
This is so simple so here...
A right angle is 90 degrees.
We learn that WXZ is 35 degrees.
90-35=55
YXZ is 55 degrees. I hope I could help!
The square of a number xx is 33. The number xx is between what two consecutive whole numbers?
I need help please I have to finish this
Answer: x=4
Step-by-step explanation:
We know that line segment AC= AB + BC
we also know that AB= 3x+8 and BC= 2x-5
by substituting these, we get AC= (3x+8)+(2x-5)
AC is given as being 23, so we know that
23= (3x+8)+(2x-5)
by combining like terms, we get that
5x+3=23
simplify this to get that x=4
Factor the expression using the GFC 70+95=
Answer:
165
Step-by-step explanation:
Answer: 165
Step-by-step explanation:
divide 8a⁸-20a⁶+2a⁴/4a⁴
-20a with reminder of 6 and Plus 17a with reminder of 2 over 2
if f(x)=2x and g(x)=5x, what is f(g(x)) when x=3
Answer:
30
Step-by-step explanation:
First: f(3)=2(3) is 3F=6, divide by 3 on each side and F=2.
Second g(3)=5(3) is 3G=15, divide by 3 on each side and G=5.
Finally, f(g)(x) is (2)(5)(3) = 10x3 = 30.
The value of the function f ( g ( x ) ) when x = 3 is 30
What is Composition of functions?
Evaluation of a function at the value of another function is known as Composition of function. A function composition is a process in which two functions, f and g, form a new function, h, in such a way that
h(x) = g(f(x)). This signifies that function g is being applied to the function x. So, in essence, a function is applied to the output of another function.
Given data ,
Let the first function be f (x ) = 2x
Let the second function g ( x ) = 5x
Now ,
The composite function is of the form f ( g ( x ) ) ,
So assign the value of x in f ( x ) as x = g ( x )
And , g ( x ) = 5x
So , the composite function will be
f ( 5x ) = 2 ( 5x )
= 10x
Now , when x = 3
f ( g ( 3 ) ) = 10 x 3
= 30
Therefore , the value of f ( g ( 3 ) ) is 30
Hence , the value of the function f ( g ( x ) ) when x = 3 is 30
To learn more about composite functions click :
https://brainly.com/question/17673639
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