Solve the system of differential equations [x' = 3x - 15y y' = 0x - 2y x(0) = 3, y(0) = 2 x(t) = 3e-2t X y(t) = e-2t
Answers
The solution to the system of differential equations is:
x(t) = 3e^(-3t),y(t) = 2e^(-2t).To solve the system of differential equations:
Start by finding the general solutions for each equation separately.
For the equation x' = 3x - 15y:
We can rewrite it as dx/dt = 3x - 15y.
This is a first-order linear homogeneous differential equation.
The general solution for x(t) can be found using the integrating factor method or by solving the characteristic equation.
Using the integrating factor method, we multiply the equation by the integrating factor e^(∫3 dt) = e^(3t) to make it integrable:
e^(3t)dx/dt - 3e^(3t)x = -15e^(3t)y.
Now, we integrate both sides with respect to t:
∫e^(3t)dx - 3∫e^(3t)x dt = -15∫e^(3t)y dt.
This simplifies to:
e^(3t)x = -15∫e^(3t)y dt + C1,
where C1 is the constant of integration.
Simplifying further:
x = -15e^(-3t)y + C1e^(-3t).
For the equation y' = 0x - 2y:
This is a separable first-order linear differential equation.
We can separate the variables and integrate both sides:
dy/y = -2dt.
Integrating both sides:
∫dy/y = -2∫dt,
ln|y| = -2t + C2,
where C2 is the constant of integration.
Taking the exponential of both sides:
|y| = e^(-2t + C2) = e^(-2t)e^(C2).
Since C2 is an arbitrary constant, we can combine it with e^(-2t) and write it as another arbitrary constant C3:
|y| = C3e^(-2t).
Considering the absolute value, we can have two cases:
Case 1: y = C3e^(-2t),
Case 2: y = -C3e^(-2t).
Now, we can use the initial conditions x(0) = 3 and y(0) = 2 to determine the specific values of the constants.
For x(0) = 3:
3 = -15e^0(2) + C1e^0,
3 = -30 + C1,
C1 = 33.
For y(0) = 2:
2 = C3e^0,
C3 = 2.
Plugging in the specific values of the constants, we obtain the particular solutions.
For x(t):
x = -15e^(-3t)y + C1e^(-3t),
x = -15e^(-3t)(2) + 33e^(-3t),
x = -30e^(-3t) + 33e^(-3t),
x = 3e^(-3t).
For y(t):
y = C3e^(-2t),
y = 2e^(-2t).
Therefore, the solution to the system of differential equations is:
x(t) = 3e^(-3t),
y(t) = 2e^(-2t).
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Related Questions
What is 4 less than the quotient of x and 3?
Answers
Answer:
Step-by-step explanation:
Quotient means divide. So you are dividing x by 3 written as x/3
Then you subtract 4
x
===== - 4
3
Solve the absolute value equation.
|x|-7 = -3
A. x = 14
B. x = +10
C. x = 4, 10
D. x = -4, -10
Answers
A. x= 4
Step-by-step explanation:
|x|-7 = -3
x = -3 +7
|x|= 4
x= 4
x=-4
find the value of x when
m<2= x + 64
Answers
Answer:
x = -9
if this is right, brainliest is much appreciated!
Step-by-step explanation:
hope this helps!
Answer: x = -9
Step-by-step explanation:
top right angle: Congruent sides implies congruent angles = ∠2
Using the Triangle Sum Theorem, the sum of the angles = 180
70° + ∠2 + ∠2 = 180°
2(∠2) = 110°
∠2 = 55°
x + 64 = 55
x = -9
Endure all, a manufacturer of batteries claims that the lifetime of their batteries is normally distributed with a mean of 500 hours and a standard deviation of 40 hours. What is the probability that an endure all battery selected at random will last more than 550 hours?.
Answers
An Endure chosen randomly the likelihood of any battery lasting longer than 550 hours is 10.56%.
What does z score mean?An indicator of how nearly a value relates to the mean of either a set of values is called a Z-score.Z-score is quantified using standard deviations relative to the mean.The data point's entire score is equal to the mean score when the Z-score is 0.The amount that the raw score departs first from mean is determined using the Z score.These steps are used to determine the Z score:
z = (x - μ)/σ
As mentioned in question-
x > 550 and μ = 500, σ = 40:
z = 550 - 500 / 40
z = 1.25
Obtain the p-value from normal distribution chart,
P(z > 1.25) = 1 - P(z < 1.25)
P(z > 1.25) = 1 - 0.8944
P(z > 1.25) = 0.1056
Thus, an Endure chosen at random the likelihood of any battery lasting longer than 550 hours is 10.56%.
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1. Use an antiderivative to show that for every contour C extending from a point z
1
to a point z
2
, ∫
C
z
n
dz=
n+1
1
(z
2
n+1
−z
1
n+1
)(n=0,1,2,…)
Answers
The formula ∫Cz^n dz = (z^(n+1))/(n+1) holds true for every contour C extending from z1 to z2, using antiderivative F(z) of z^n as F(z) = (z^(n+1))/(n+1) + C.
Let's consider the contour integral of z^n with respect to z over a contour C extending from z1 to z2. We can use the Fundamental Theorem of Calculus to evaluate this integral by finding an antiderivative of z^n.
We have the antiderivative F(z) of z^n as F(z) = (z^(n+1))/(n+1) + C, where C is a constant of integration.
Now, applying the Fundamental Theorem of Calculus, we can evaluate the integral ∫Cz^n dz by subtracting the value of F(z) at the endpoints:
∫Cz^n dz = F(z2) - F(z1) = [(z2)^(n+1)/(n+1) + C] - [(z1)^(n+1)/(n+1) + C]
= (z2^(n+1) - z1^(n+1))/(n+1)
Therefore, we have shown that for every contour C extending from z1 to z2, the integral of z^n with respect to z is given by (z2^(n+1) - z1^(n+1))/(n+1). This formula holds true for any nonnegative integer n.
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I need help on this problem..please and thanks u
Answers
Answer:
A is on a line with the slope of 2/3
Step-by-step explanation:
the rule is that you do y2-y1 over x2-x1 which you do 4-2 over 8-5 and your final answer you would get 2/3.
hope this helped!
i am new here and my name is arrianna call me arri for short
Answer:
a
Step-by-step explanation:
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.
r ≥ 2, π ≤ θ ≤ 2π
Answers
The resulting region in the plane is a shaded annulus centered at the origin, with inner radius 2 and outer radius infinity.
The region in the plane consists of all points with polar coordinates (r, θ) such that r is greater than or equal to 2, and θ is between π and 2π.
To sketch this region, we can start by drawing the circle with radius 2 centered at the origin. This circle corresponds to the boundary r=2 of the region.
Next, we shade in the region to the right of the vertical line passing through the origin, since this corresponds to the interval π ≤ θ ≤ 2π.
Finally, we shade in the interior of the circle, since we want to include all points with r greater than or equal to 2.
The resulting region in the plane is a shaded annulus centered at the origin, with inner radius 2 and outer radius infinity. The boundary of the region is the circle of radius 2 centered at the origin, together with the positive x-axis.
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A family buys 6 tickets to a show. They also each spend $3 on a snack
Answers
Answer:
6*3=18
....
....
....
....
...
...
Anyone have any idea how to do this?
Answers
Answer:
does my answer sheet help
Step-by-step explanation:
if it does not feel free to report me :)
Help me please I am having trouble figuring out the answer. Help me find the ratio.
Answers
Answer:
not equivalent to meteorologists ratio
Step-by-step explanation:
meteorologists ratio is
rainy days : sunny days = 2 : 5
last months weather is
rainy days : sunny days
= 10 : 20 ( divide both parts by LCM of 10 )
= 1 : 2 ← not equivalent to 2 : 5
helpppppppppppppppppppppppppppppp meeeeeeeeeeeeeeeeeee plsssssssssssssssssssssss there is more but plss help mee
Answers
Find the value of the variable
125
Answers
The angles that measures 125 and y make a linear pair (equal 180)
So you just do 180-125 to get y
HELP PLSPLSPSL check image
Answers
Answer:
D
Step-by-step explanation:
HELP!
Please help me!
Answers
Answer:I think it's C although I might be wrong
Step-by-step explanation:
Answer: B
Hope this helps :)
HELP PLEASE!! the zero of a quadratic function is x=-1/2. what does this tell you about the graph of this quadratic function?
Answers
The correct statement about the parabola is,
⇒ the parabola crosses the x-axis at x=- 1/2 and x=1/2
Since, ax² +bx +c = 0 is an quadratic equation its zeroes are the values of x for which the Left hand side becomes zero.
To find zeroes we can use quadratic formula that is
x = (-b ±√(b²-4ac))/2
Graph of the quadratic equation are points ( x , y(x) )
where y(x) = ax² +bx +c
when x= t is zero of quadratic equation ax² +bx +c = 0
then y(t) = 0
(t, 0) points are points on the x-axis.
We have to given that;
The zero of a quadratic function is x=-1/2.
Hence, We can conclude that;
The zero of a quadratic function is x=1/2
Thus, The correct statement about the parabola is,
⇒ the parabola crosses the x-axis at x=- 1/2 and x=1/2
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please answer correctly !!!!! Will mark Brianliest !!!!!!!!!!!!!!
Answers
Answer:
56.443°
Step-by-step explanation:
sin b= 5/6
b= sin-¹ (5/6) = 56.443°
4. Which is NOT a way in which thermal energy is transferred? A. Conduction , B. Convection , C. Radiation , D. Sublimation
WILL MARK BRAINLIEST!!
Answers
Answer:
Convection
Step-by-step explanation:
Conduction means the heat energy is being transferred .Hence its not the ansRadiation is a process happened in radioactive elements like radium,uranium etc by which they release beta or alpha particles.Sublimation happens like a solid converted to gas or gas converted to solid directly.The correct option is C Radiation.
Conduction means transfer of energy (heat) arising from temperature differences between adjacent parts of a body.
Convection is the transfer of thermal energy by the physical movement of fluid (liquid, gas, or plasma) from one location to another.
Radiation is an electromagnetic radiation that is emitted by a body as a result of its temperature. All objects with a temperature above absolute zero emit thermal radiation in a spectrum of wavelength.
Sublimation is the process of changing a solid into a gas without passing through the liquid phase. To sublime a substance, a certain energy must be transferred to the substance via heat (q) or work (w).
Hence from all the above definitions it is clear that every phenomenon except radiation transfer some amount of thermal energy.
Hence through Radiation thermal energy is not transferred.
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Find the slope of the line that passes through
(10, 1) and (5,2)
Answers
Answer:
the slope of the line is -1/5
remember, rise over run.
Step-by-step explanation:
May I have brainliest please? :)
Answer:
\(m=-\frac{1}{5}\)
Step-by-step explanation:
3/5 + 1/2 in simplest form
Answers
3/5 = 6/10
1/2 = 5/10
6/10 + 5/10 = 11/10 = 1 1/10
Simplified it would be \(\frac{11}{10}\).
Wyatt is going to invest $58,000 and leave it in an account for 19 years. Assuming the interest is compounded continuously, what interest rate, to the nearest tenth of a percent, would be required in order for Wyatt to end up with $109,000?
The answer which is the rate, should be 3.3%
3.3%
Round to the nearest tenth of a percent which is 3.3 %
Answers
Answer:
Step-by-step explanation:
3.3%
Complete the following statement. Use the integers that are closest to the number in the middle. < 82 <
Answers
The integers that are closest to the number in the middle in the inequality x < √82 < y are x = 9 and y = 11.
To find the integers x and y that satisfy the inequality x < √82 < y, we need to find the two integers that are closest to the square root of 82.
First, we know that the perfect square that is less than 82 is 81, which is equal to 9². Therefore, we can say that 9 < √82.
Next, we need to find the next closest integer to the square root of 82. To do this, we can calculate the square of 10 and the square of 11, which are 100 and 121, respectively. Since 82 is closer to 81 than to 100, we can say that √82 < 10.5.
Therefore, x = 9 and y = 11 are the integers that satisfy the inequality x < √82 < y.
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The question is -
Complete the following statement. Use the integers that are closest to the number in the middle. x < √82 < y, find x and y.
the national health and nutrition examination survey (nhanes) reported that in a recent year, the mean serum cholesterol level for u.s. adults was 202, with a standard deviation of 41 (the units are milligrams per deciliter). a random sample of 100 adults is chosen. what is the probability that the sample mean cholesterol level is less than 190?
Answers
Therefore, the probability that the sample mean cholesterol level is less than 190 is approximately 0.23%.
We can use the central limit theorem to approximate the distribution of the sample mean cholesterol level as normal with a mean of 202 and a standard deviation of 41/sqrt(100) = 4.1.
To find the probability that the sample mean cholesterol level is less than 190, we can standardize the sample mean using the formula:
z = (x - mu) / (sigma / sqrt(n))
where x is the sample mean, mu is the population mean, sigma is the population standard deviation, and n is the sample size.
Plugging in the values, we get:
z = (190 - 202) / (4.1) = -2.93
Now we can use a standard normal distribution table or calculator to find the probability that a standard normal random variable is less than -2.93. The probability is approximately 0.0023 or 0.23%.
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what is the answer to this equation ... 3 + 2c + 5c + 2 ??
Answers
you can’t add the c and the regular number unless they have the same variable
Answer:
solution:7c+5
I need help with this math
Answers
Answer:
50 i hope your helpful
......
Answer:
it is 50
Step-by-step explanation:
i really hope this helped bestie!
the length of a rectangle is 6 more than twice the width. the perimeter is 150 meters. find the length and width.
Answers
Answer:
The length is 52 meters and the width is 23 meters.
Step-by-step explanation:
Let l stand for length, and w stand for width.
l = 2w + 6
The length is 6 more than twice the width.
2l + 2w = 150
Twice the length plus twice the width is the perimeter of a rectangle.
2(2w + 6) + 2w = 150
4w + 12 + 2w = 150
6w + 12 = 150
6w = 138
w = 23
l = 2(23) + 6
l = 46 + 6
l = 52
6. daniel's age is 7 more than three times his
daughter lucy's age. the difference between
twice daniel's age and five times lucy's age is
24. find the ages of daniel and lucy.
help pls
Answers
Answer:
Lucy: 10
Daniel: 37
Step-by-step explanation:
Let Daniel's age be d, and Lucy's age be l.
We can set up a system of equations.
d = 7 + 3l
2d-5l = 24
We can subsitute the value of d from the first equation into the second equation.
2*(7+3l)-5l = 24
Distribute:
14 + 6l - 5l = 24
Combine like terms:
l = 10
Plug this value back into the first equation:
d = 7 + 3*10
d = 37
¿Cuál es la Razón trigonométrica del ángulo notable de 45 grados?
Answers
Answer:
Solutions
Step-by-step explanation:
El ángulo de 45 grados es uno de los ángulos notables en trigonometría. La razón trigonométrica más comúnmente asociada con este ángulo es:
- La razón trigonométrica del seno: sin(45°) = 1/√2 ≈ 0.7071
- La razón trigonométrica del coseno: cos(45°) = 1/√2 ≈ 0.7071
- La razón trigonométrica de la tangente: tan(45°) = sin(45°)/cos(45°) = 1
Recuerda que estas razones trigonométricas se obtienen a partir de la relación entre los lados de un triángulo rectángulo y se utilizan para calcular longitudes, ángulos y relaciones en problemas trigonométricos.
Enter the mixed number as a decimal.
9 3/5
Answers
Answer:
9.6
Step-by-step explanation:
bc 3/5 = 0.6
hope it helps
Answer:
9.6
Step-by-step explanation:
1/5=20%
Please help me with this question!!!!!
Answers
h = 11.9 cm
cos = adjacent/ hypotenuse
therefore:
cos(24) = h/ 13
rearrange:
h = 13cos(24)
put into calculator:
h = 11.8760...
rounded to one decimal point:
h = 11.9cm
Kim sells newspapers. she is paid $10 per week and $0.18 per news paper sold one week she sells 829 newspapers how much does she she earn selling newspapers that week?
Answers
Kim earned \($159.22\) for the week by selling 829 newspapers.
Kim's earnings for the week, we need to use the information provided in the problem.
She is paid a base salary of \($10\) per week, and she also earns \($0.18\) for each newspaper she sells.
She earned for selling 829 newspapers, we need to multiply the number of newspapers by the amount she earns per newspaper, and then add her base salary:
Earnings from newspapers sold = 829 × \($0.18\)
= \($149.22\)
Earnings for the week = \($149.22 + $10\)
=\($159.22\)
It's worth noting that if Kim didn't sell any newspapers, her earnings for the week would still be \($10\), which is her base salary.
She will always have some income even if she has a slow week and doesn't sell many newspapers.
The more newspapers she sells, the higher her total earnings will be.
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