Find the general solution of the given second-order differential equation. 25y'' − 15y' − 4y = 0
Answers
Final answer:
To solve the second-order differential equation 25y'' - 15y' - 4y = 0, we find the characteristic equation and its roots, then express the general solution as a linear combination of terms with the found roots as exponents, resulting in [tex]y(t) = C_1e^{-t/5} + C_2e^{4t/5}.[/tex]
Explanation:
We are given a second-order differential equation with constant coefficients: 25y'' − 15y' − 4y = 0. First, we will look for a characteristic equation that arises from this differential equation to find the general solution.
To find the characteristic equation, we substitute [tex]y = e^{rt}[/tex] (where r is a constant) into our differential equation, replacing y'' with [tex]r^2e^{rt}[/tex], y' with [tex]re^{rt}[/tex], and y with [tex]e^{rt}[/tex]. The equation becomes 25r² − 15r − 4 = 0.
Solving for r, we find the roots of the characteristic equation, which are the values of r that satisfy the equation.
Once we have the roots, which we'll call r₁ and r₂, the general solution to the differential equation is [tex]y = C_1e^{r_1t} + C_2e^{r_2t}[/tex], where C₁ and C₂ are arbitrary constants determined by initial conditions.
To find r₁ and r₂, we factor the characteristic equation:
25r² − 15r − 4 = (5r + 1)(5r - 4) = 0. Hence, r₁ = -1/5 and r₂ = 4/5.
Finally, the general solution to the differential equation is [tex]y(t) = C_1e^{-t/5} + C_2e^{4t/5}.[/tex]
Related Questions
A rectangular pool is 12m long and 6m wide. A walkway of uniform width surrounds the pool. If the area of the walkway is 9m squared less than that of the pool, how wide is the walkway?
Answers
The area of the pool is 12(6) = 72 m².
Adding a walkway of unknown width, x, around all 4 sides of the pool increases the width by 2x and the length by 2x; thus the area of the entire pool and walkway together would be given by
(12+2x)(6+2x)
We know that the area of just the walkway is 9 m² less than the area of the pool. This means that:
(12+2x)(6+2x)-72 = 72-9
Multiplying through we have:
12*6+12*2x+2x*6+2x*2x - 72 = 63
72 + 24x + 12x + 4x² - 72 = 63
24x + 12x + 4x² = 63
36x + 4x² = 63
Writing in standard form we have:
4x² + 36x = 63
We want to set it equal to 0 to solve, so subtract 63 from both sides:
4x² + 36x - 63 = 63 - 63
4x² + 36x - 63 = 0
Using the quadratic formula,
[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \\=\frac{-36\pm \sqrt{36^2-4(4)(-63)}}{2(4)} \\ \\=\frac{-36\pm \sqrt{1296--1008}}{8}=\frac{-36\pm \sqrt{1296+1008}}{8} \\ \\=\frac{-36\pm \sqrt{2304}}{8}=\frac{-36\pm 48}{8}=\frac{-36+48}{8}\text{ or }\frac{-36-48}{8} \\ \\=\frac{12}{8} \text{ or }\frac{-84}{8}=1.5 \text{ or }-10.5[/tex]
Since a negative width makes no sense, the walkway is 1.5 m wide.
At Central High School, there are 60 football players and 25 cheerleaders. What is the ratio of cheerleaders to football players? please slimlfy
Answers
Answer:
The answer is 5:12
Step-by-step explanation:
PLZ HELP
The price of a technology stock was 9.83
yesterday. Today, the price fell to 9.72
. Find the percentage decrease. Round your answer to the nearest tenth of a percent.
@nilbert
Answers
percent change = ((new value)/(original value) - 1) × 100%
For your numbers, that is
percent change = ((9.72/9.83) - 1) × 100% = -0.01119... × 100%
The decrease in value is about 1.12%
Can someone help me please!!
Answers
When we factor out the GCF of -3, we get:
-3(q^2 + 7q + 10)
Next, we must look for two numbers that will serve as coefficients for the variables in factored form. These two numbers must add to 7 (the middle term) and multiply to 10 (the constant or last term). The two numbers that satisfy these parameters are 5 and 2.
Therefore, we must use these numbers to create factors of the innermost factor that we have left, to get:
-3(q+5)(q+2)
The above expression is your final answer, because you cannot factor any more. And remember: you can always check to make sure that your factoring is correct by multiplying out the factors and making sure you get your original trinomial.
I hope this helps!
FIND.......16 2/3 % of 96 lamps
Answers
16 2/3 = 16 2/3 ÷ 100 = 1/6
16 2/3 % of 96 = 1/6 x 96 = 16
Ans: 16 2/3 % of 96 lamps = 16
the angles formed by two perpendicular lines have a measure of
Answers
Angles of perpendicular lines are always 90 degrees.
Hope this helps!
32% of 50 please answer fast need help ............
Answers
the answer is 16.
32% needs to change to 0.32. then, you multiply 0.32 by 50, and the sum is 16.
hope you have a great day / night :)
Tom and Sonya were comparing their answers on a math problem. It was an order of operations problem, and they each had a different answer. Tom had an answer of 11, and Sonya had an answer of 104. Which student correctly solved the order of operations problem given by the teacher?
(52 +6)–(7•4)+8
*Challenge: Can you nd the mistake that Sonya made in completing her problem?
Answers
= (52 + 6) - (7 * 4) + 8
work inside parentheses first
= 58 - 28 + 8
work left to right now
= 30 + 8
= 38
SONYA'S MISTAKE
= (52 + 6) - (7 * 4) + 8
I think she added the 52, 6+4, multiplied 7*4 to equal 28, then added the 6+8 to equal 14
= 52 + 28 + 14 + 10
= 104
ANSWER: 38
Hope this helps! :)
the average number of minutes that it took jermaine to complete his homework assignments for a week in september was 27 minutes, with a mean absolute deviation of 4 minutes. with this informatiom, which statement is true. Pls someone help me
Answers
Answer:
On average, Jermaine spent between 23 and 31 minutes on his homework.
On average, Jermaine spent between 23 and 31 minutes on his homework.
What is Mean absolute deviation?This is defined as the average of the absolute deviations from a central point.
In this scenario, the mean absolute deviation
= 27 ± 4 minutes
= 27+4 or 27 - 4
= 31 or 23 minutes.
Therefore we can infer that Jermaine spent between 23 and 31 minutes on his homework.
Read more about Mean absolute deviation here https://brainly.com/question/447169
Mark’s salary is $900 per month. What is his yearly salary? $
Answers
What is the approximate volume of a cylinder with a radius of 2 centimeters and height of 10 centimeters? Drag and drop the answer into the box.
A. 125.6
B. 251.3
C. 502.7
D. 1005.3
Answers
V = pi * r^2 h
where
V = ??
pi = 3.14
r = 2 cm
h = 10 cm
V = 3.14 * 2*2 * 10
V = 3.14 * 4 * 10
V = 125.6 cm^3
Choice A <<<<====
Answer:
125.6
Step-by-step explanation:
Jean adds 35 and 9. How can she solve using only equations to model her thinking?
Answers
35+9=44
Or she may want to break up the two digit number into two or more parts, for example: (5×6) + (1×5) + 9 = 44 or (7×5) + 9 = 44
Alice says she can find the area of the room by dividing the floor plan into two trapezoids. show how she can divide the floor plan. then find the area using her method
Answers
To find the room's area, Alice can divide the floor plan into two trapezoids, calculate each trapezoid's area using the formula for a trapezoid's area (A = (a + b)/2 × h), and then sum the two areas.
Alice can find the area of the room by first dividing the floor plan into two trapezoids, which can be done by drawing a line across the room that connects two non-adjacent vertices. Once divided, you can use the formula for the area of a trapezoid to find the area of each part and then sum the two areas to get the total area of the room.
Calculating the Area of a Trapezoid
The formula for the area of a trapezoid is:
A = (a + b)/2 × h
Where a and b are the lengths of the two parallel sides, and h is the height. Once you calculate the area of both trapezoids using this formula, you can add them together to find the total area of the room.
Number of dinners = 125 Individual portion = 8 oz. Size of can = 32 oz. Number of cans required = _____. 2
Answers
125 * 8 = 1000 oz
To get number of cans:
1000 ÷ 32 = 3.125 cans or at least 4 cans to served dinner
Answer:
Number of cans required = 32
Step-by-step explanation:
Number of dinners = 125
Individual portion size of each dinner = 8 oz.
Total portion size of all the dinners = 125 × 8
⇒ Total portion size of all the dinners = 1000 oz.
Size of one can = 32 oz.
[tex]\text{Number of cans required = }\frac{\text{Total portion size of dinner}}{\text{Size of each can}}\\\\\implies\text{Number of cans required = }\frac{1000}{32}\approx 31.25[/tex]
Hence, Number of cans required = 32
which of the following must be true?
check ALL that apply,
Answers
_____
The Law of Cosines tells you
.. c^2 = a^2 +b^2 -2ab*cos(θ)
Your problem statement provides a relation involving c^2, so we can substitute to get
.. a^2 +b^2 > a^2 +b^2 -2ab*cos(θ)
.. 0 > -2ab*cos(θ)
.. cos(θ) > 0
In the range 0 ≤ θ ≤ 180°, cos(θ) will be positive for θ < 90°, an acute angle.
Thus A and C are false, B and D are true.
What is the value of X?
Write as decimal.
Answers
dMN/dMP=dAM/x
dAM=46.2-14=32.2 cm
then
x=dAM*[dMP/dMN]=32.2*[72.6/46.2]=50.60 cm
the answer is x=50.60 cm
what is the probability of randomly picking a girl without brown hair
Answers
Express the number .0957 as a percent rounded to the hundredth place. (please include the % symbol.)
Answers
A cell phone uses a 3.0 V battery. The circuit board it uses needs a 0.05 A current. What size resistor is needed to generate this current?
___ Ω
Answers
V = IR
where V - voltage
I - current passing
R - resistance
voltage is the potential difference between 2 points and current passing through these 2 points is proportional to the potential difference, in other words voltage.
V∞I
Resistance is the constant in the equation
the current required to pass through is 0.05 A (amperes)
the battery voltage is 3.0 V (volts)
substituting these values
3 V = 0.05 A * R
R = 3/0.05
R = 60 Ω (ohms)
Answer:
the answer is 60.
Step-by-step explanation:
Which of the following uses absolute value correctly to show the distance between -40 and 18
Answers
|-40 -18| = |-58| = 58 . . . . units
_____
In the first selection, it is trying to tell you |-58| = -58. That is not so. The result of taking the absolute value is that the sign is never negative.
if 100 envelopes cost 70 cents, how much would 250 cost?
Answers
PLZ HELP.ONLY ANSWER IF YOUR 100% SURE. which equation represents a non proportional situation? A. y = 4x B. y = 5x - 1 C. y = 16x D. y = -8x
Answers
-1 is a y-intercept
if there is a y-intercept, it is non proportional
Hope I helped
Let me know if you need anything else!
~ Zoe
Which is a better buy? 20-ounce box of cereal for $3.60 or 28 ounce for $4.20?
Answers
PRODUCT 1:
=$3.60 ÷ 20
= $0.18 per ounce
PRODUCT 2:
= $4.20 ÷ 28
= $0.15 per ounce
ANSWER: The better buy is product two at $0.15 an ounce.
Hope this helps! :)
math teacher doesn't explain this very well, please help solve.
Answers
[tex]P_s = 4s/tex]
[tex]P_t = 3s=s+s+s[/tex]
Now, we can set up our equation:
[tex]4(x-3) = 8 + x + x[/tex]
[tex]4x-12 = 8+2x[/tex]
We can solve for x:
[tex]2x-12=8[/tex]
[tex]2x = 20[/tex]
[tex]x = \frac{20}{2} = 10[/tex]
So, x = 10.
b) We known the perimeter formulas and we have what x is. So, let's just plug and chug.
Square:
[tex]4(x-3) = 4(10-3) = 40-12 = 28[/tex]
So, the perimeter of the square is 28 units².
Since their perimeters are equal the perimeter of the triangle is also 28 units²
Evaluate the expression 28C14
Answers
xCy = x! / (x-y)! y!
! means factorial
(28 x 27 x 26...x 3 x 2 x 1)/(14 x 13 x 12...x 3 x 2 x 1)(14 x 13 x 12...x 3 x 2 x 1) = 40116600
an investment is expected to pay 5 per year compounded continuously. If you want the value of the investment to be $700,000 after 25 years, how much should you invest initially? Round to the nearest dollar
Answers
700,000 = P·e^(0.05·25) ≈ 3.490343P
Then
700,000/3.490343 = P
You should invest $200,553 initially.
About 70% of Earth's surface is covered by water. If the diameter of earth is about 12,750km, find the area not covered by water to the nearest 100,000km^2.
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write an expression that is,equivalent tp 7/8
Answers
[tex] \frac{7}{8} [/tex] = [tex] \frac{3.5}{4} [/tex]
Enjoy.
~Isabella
Savanna had 40 iris blooms last year. This year she had 15% more iris blooms. How many more iris blooms did Savanna have this year?
Answers
40+15%=46
hope one of these helped
40 iris blooms -----> 100%
x iris blooms -------> 115%
Clearing the value of x we have:
x = (115/100) * (40)
x = 46
Then, comparing with last year we have:
40-46 = 6
Answer:
Savanna had this year 6 more iris blooms than last year.
what is the surface area of the following figure and I need work to please and thank you