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4 years ago

Find the general solution of the given higher-order differential equation. d3u dt3 + d2u dt2 − 2u = 0

Mathematics

1 answer:

STALIN [3.7K]4 years ago

6 0

$\dfrac{\mathrm d^3u}{\mathrm dt^3}+\dfrac{\mathrm d^2u}{\mathrm dt^2}-2u=0$

This ODE has characteristic equation

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

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

which has roots at $r=1,-1\pm i$ . Then the characteristic solution to the ODE is

$u(t)=C_1e^t+C_2e^{(-1+i)t}+C_3e^{(-1-i)t}$

$\implies\boxed{u(t)=C_1e^t+C_2e^{-t}\cos t+C_3e^{-t}\sin t}$

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The thickness of a plastic film (in mils) on a substrate material is thought to be influenced by the temperature at which the co

statuscvo [17]

Answer:

Step-by-step explanation:

Hello!

The objective is to test if rising the process temperature reduces the thickness of the plastic film that coats a substrate material To do so, two samples of substrates are coated at different temperatures:

Sample 1

X₁: Thickness of the plastic film after the substrate is coated at 125F

n₁=11

X[bar]₁= 101.28

S₁= 5.08

Sample 2

X₂: Thickness of the plastic film after the substrate is coated at 150F

n₂= 13

X[bar]₂= 101.70

S₂= 20.15

Does the data support this claim? Use the P-value approach and assume that the two population standard deviations are not equal.

Now if the higher the heat, the thinner the thickness of the plastic coating, then the average thickness of the coating done at 150F should be less than the average thickness of the coating done at 125F, symbolically: μ₂ < μ₁

Then the hypotheses are:

H₀: μ₂ ≥ μ₁

H₁: μ₂ < μ₁

α:0.05 (there is no α level stated so I've chosen the most common one)

Assuming that both variables have a normal distribution since the population standard deviations are not equal, the statistic to use is the Welch's t-test:

$t= \frac{(X[bar]_2-X[bar]_1)-(Mu_1-Mu_2)}{\sqrt{\frac{S^2_1}{n_1} +\frac{S^2_2}{n_2} } } ~~t_w$

$t_{H_0}= \frac{(101.7-101.28)-0}{\sqrt{\frac{(20.15)^2}{13} +\frac{(5.08)^2}{11} } } = 0.072$

This test is one-tailed to the left, meaning that you'll reject the null hypothesis at small values of t. The p-value is also one-tailed and has the same direction as the test. To calculate it you have to first calculate the degrees of freedom of the Welch's t:

$Df_w= \frac{(\frac{S^2_1}{n_1} +\frac{S^2_2}{n_2} )^2}{\frac{(\frac{S^2_1}{n_1})^2 }{n_1-1}+\frac{(\frac{S^2_2}{n_2} )^2}{n_2-1} }$

$Df_w= \frac{(\frac{5.08^2}{11} +\frac{20.15^2}{13} )^2}{\frac{(\frac{5.08^2}{11})^2 }{10} +\frac{(\frac{20.15^2}{13} )^2}{12} } = 13.78$

The distribution is a Student's t with 13 degrees of freedom, then you can calculate the p-value as:

P(t₁₃≤0.072)= 0.4718

Using the p-value approach, the decision rule is:

If the p-value ≤ α, the decision is to reject the null hypothesis.

If the p-value > α, the decision is to not reject the null hypothesis.

The p-value is greater than the significance level, so the decision is to nor reject the null hypothesis.

Using a significance level of 5%, there is no significant evidence to support the claim that the average thickness pf the plastic coat processed at 150F is less than the average thickness pf the plastic coat processed at 125F.

I hope you have a SUPER day!

4 0

3 years ago

Find the product. Your answer must be in simplest form. 7/8 of 56

Leviafan [203]

Answer:

You probably know that the number above the fraction line is called the numerator and the number below it is called the denominator. To work out the fraction of any number, we first need to convert that whole number into a fraction as well.

Here's a little tip for you. Any number can be converted to fraction if you use 1 as the denominator:

So now that we've converted 56 into a fraction, to work out the answer, we put the fraction 7/8 side by side with our new fraction, 56/1 so that we can multiply those two fractions.

That's right, all you need to do is convert the whole number to a fraction and then multiply the numerators and denominators. Let's take a look:

7 x 56

8 x 1

392

In this case, our new fraction can actually be simplified down further. To do that, we need to find the greatest common factor of both numbers.

You can use our handy GCF calculator to work this out yourself if you want to. We already did that, and the GCF of 392 and 8 is 8.

We can now divide both the new numerator and the denominator by 8 to simplify this fraction down to its lowest terms.

392/8 = 49

8/8 = 1

When we put that together, we can see that our complete answer is:

The complete and simplified answer to the question what is 7/8 of 56 is:

Hopefully this tutorial has helped you to understand how to find the fraction of any whole number. You can now go give it a go with more numbers to practice your newfound fraction skills.

Step-by-step explanation:

5 0

3 years ago

Read 2 more answers

Determine the slope of the line that passes through each pair of points. A (0,1) ,B (2,7)

valkas [14]

Your answer should be 3

Use this formula to solve slopes:

m = (y2 – y1) / (x2 – x1)

8 0

3 years ago

Which polynomial lists the powers in descending order? A. –10 + 4x3 – 4x5 + 2x4 + x7 B. x7 – 4x5 + 2x4 + 4x3 – 10 C. x7 + 4x3 +

Artist 52 [7]

The answer for the exercise shown above is the option B, which is: $x^{7} -4 x^{5} +2 x^{4} +4 x^{3}- 10$
The explanation is shown below:
To list the powers of the polynomial given in the problem above in descending order, you must write the terms from the highest degree to the lowest degree. You have that the highest degree is $7
$ and the lowest degree is $0
$ (Because $-10 x^{0} =-10(1)=-10$ . Therefore, you have that the correct option is the option B.

4 0

3 years ago

Chuck's Computer Corporation has 10,000 shares of stock outstanding.

cupoosta [38]

Hi there
the dividend per share of stock
15,000÷10,000
=1.5 per share
if you owned 50 shares
1.5×50
=75

Hope it helps

8 0

4 years ago

Read 2 more answers