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3 years ago

How many significant figures are in the number 3,255.4

Mathematics

1 answer:

Charra [1.4K]3 years ago

7 0

Answer:

Number of Significant Figures: 5

The Significant Figures are 3 2 5 5 4

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Solve the given DE<br><br> dx/dt=3x+2y+1<br><br> dy/dt=-2x-y+1

kirill115 [55]

We can solve for $x(t)$ first by rewriting the system of first-order ODEs as a single second-order ODE in $x(t)$ :

Taking the derivative of the first ODE gives

$\dfrac{\mathrm dx}{\mathrm dt}=3x+2y+1\implies\dfrac{\mathrm d^2x}{\mathrm dt^2}=3\dfrac{\mathrm dx}{\mathrm dt}+2\dfrac{\mathrm dy}{\mathrm dt}$

while solving for $2y$ gives

$\dfrac{\mathrm dx}{\mathrm dt}=3x+2y+1\implies2y=\dfrac{\mathrm dx}{\mathrm dt}-3x-1$

Then

$\dfrac{\mathrm d^2x}{\mathrm dt^2}=3\dfrac{\mathrm dx}{\mathrm dt}+2(-2x-y+1)$

$\dfrac{\mathrm d^2x}{\mathrm dt^2}=3\dfrac{\mathrm dx}{\mathrm dt}-4x-2y+2$

$\dfrac{\mathrm d^2x}{\mathrm dt^2}=3\dfrac{\mathrm dx}{\mathrm dt}-4x-\left(\dfrac{\mathrm dx}{\mathrm dt}-3x-1\right)+2$

$\implies\dfrac{\mathrm d^2x}{\mathrm dt^2}-2\dfrac{\mathrm dx}{\mathrm dt}+x=3$

which is linear with constant coefficients, so it's trivial to solve; the corresponding homogeneous ODE

$x''-2x'+x=0$

has characteristic equation

$r^2-2r+1=(r-1)^2=0$

with root $r=1$ (multiplicity 2), so the characteristic solution is

$x_c=C_1e^t+C_2te^t$

For the non-homogeneous ODE, assume a particular solution of the form

$x_p=a\implies{x_p}'={x_p}''=0$

Substituting these into the ODE gives

$0-2\cdot0+a=3\implies a=3$

Then the general solution for $x(t)$ is

$\boxed{x(t)=C_1e^t+C_2te^t+3}$

From here, we find

$\dfrac{\mathrm dx}{\mathrm dt}=C_1e^t+C_2(t+1)e^t$

so that

$2y=(C_1e^t+C_2(t+1)e^t)-3(C_1e^t+C_2te^t+3)-1$

$\implies\boxed{y(t)=\left(\dfrac{C_2}2-C_1\right)e^t-C_2te^t-5}$

3 0

4 years ago

Use the method of undetermined coefficients to find the general solution to the de y′′−3y′ 2y=ex e2x e−x

djverab [1.8K]

I'll assume the ODE is

$y'' - 3y' + 2y = e^x + e^{2x} + e^{-x}$

Solve the homogeneous ODE,

$y'' - 3y' + 2y = 0$

The characteristic equation

$r^2 - 3r + 2 = (r - 1) (r - 2) = 0$

has roots at $r=1$ and $r=2$ . Then the characteristic solution is

$y = C_1 e^x + C_2 e^{2x}$

For nonhomogeneous ODE (1),

$y'' - 3y' + 2y = e^x$

consider the ansatz particular solution

$y = axe^x \implies y' = a(x+1) e^x \implies y'' = a(x+2) e^x$

Substituting this into (1) gives

$a(x+2) e^x - 3 a (x+1) e^x + 2ax e^x = e^x \implies a = -1$

For the nonhomogeneous ODE (2),

$y'' - 3y' + 2y = e^{2x}$

take the ansatz

$y = bxe^{2x} \implies y' = b(2x+1) e^{2x} \implies y'' = b(4x+4) e^{2x}$

Substitute (2) into the ODE to get

$b(4x+4) e^{2x} - 3b(2x+1)e^{2x} + 2bxe^{2x} = e^{2x} \implies b=1$

Lastly, for the nonhomogeneous ODE (3)

$y'' - 3y' + 2y = e^{-x}$

take the ansatz

$y = ce^{-x} \implies y' = -ce^{-x} \implies y'' = ce^{-x}$

and solve for $c$ .

$ce^{-x} + 3ce^{-x} + 2ce^{-x} = e^{-x} \implies c = \dfrac16$

Then the general solution to the ODE is

$\boxed{y = C_1 e^x + C_2 e^{2x} - xe^x + xe^{2x} + \dfrac16 e^{-x}}$

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2 years ago

The Porter family used an online family budget estimator to compare the costs of living in Baltimore and San Francisco for a fam

alexandr402 [8]

The cost of living in San Francisco, California is greater than the cost of living in Baltimore.

Given that

The Porter family used an online family budget estimator to compare the costs of living in Baltimore and San Francisco for a family with two parents and two children.

We have to determine

Where is the cost of living higher?

According to the question

Cost of living is the amount of money needed for fixed expenses such as housing, food, taxes, and health care, as well as variable expenses.

Big cities tend to have a higher cost of living than smaller cities.

From a business perspective, we need to consider the cost of living and rent index (considering they will rent in both cities) difference in response to net earnings.

According to recent statistics, the cost of living in Baltimore is about 40 percent lower than in San Francisco.

Also, it should be noted the Porter Family would have to earn quite more than they do in Baltimore before they can maintain the same standard of living.

Supposing Porter's family is earning $50,000 in Baltimore, they would need about $89,000 to maintain the same lifestyle they lived in Baltimore.

Per data in general, we can say that it would cost ~$5000+ to live in California.

These values should be directly related to the indices difference of these underlying factors: consumer prices (CA is 4% lower than MD), consumer prices + rent (CA is 7% higher than MD), rent (CA is 28% higher than MD), restaurant (CA is 3% lower than MD), groceries (CA is 4% lower than MD), and local purchasing power is 10% higher in CA than MD.

Hence, the cost of living in San Francisco, California is greater than the cost of living in Baltimore, Maryland.

To know more about Statices click the link given below.

brainly.com/question/2950660

7 0

3 years ago

. A stadium sold 33,300 tickets to a concert. Which statement about this number is true?

Kobotan [32]

Answer: D ; The value of the digit in the ten thousand place is 1/10 the value of the digit in the hundreds place.

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3 years ago

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