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Alborosie
3 years ago
5

Solve the given initial-value problem.

Mathematics
1 answer:
jeka943 years ago
3 0

For the corresponding homogeneous ODE,

y'''-2y''+y'=0

the characteristic equation is

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

which admits the characteristic solution,

y_c=C_1+C_2e^x+C_3xe^x

Assume a particular solution of the form

y_p=ax+bx^2e^x+ce^{5x}

(ax because a constant solution is already accounted for by C_1; x^2e^x because both e^x and xe^x are accounted for)

\implies{y_p}'=a+(2x+x^2)be^x+5ce^{5x}

\implies{y_p}''=(2+4x+x^2)be^x+25ce^{5x}

\implies{y_p}'''=(6+6x+x^2)be^x+125ce^{5x}

Substituting the derivatives of y_p into the ODE gives

((6+6x+x^2)be^x+125ce^{5x})-2((2+4x+x^2)be^x+25ce^{5x})+(a+(2x+x^2)be^x+5ce^{5x})=2-24e^x+40e^{5x}

a+2be^x+80ce^{5x}=2-24e^x+40e^{5x}

\implies\begin{cases}a=2\\2b=-24\\80c=40\end{cases}\implies a=2,b=-12,c=\dfrac12

So the particular solution is

y=C_1+C_2e^x+C_3xe^x+2x-12x^2e^x+\dfrac12e^{5x}

With the given initial conditions, we find

y(0)=\dfrac12=C_1+C_2+\dfrac12\implies C_1+C_2=0

y'(0)=\dfrac52=C_2+C_3+2+\dfrac52\implies C_2+C_3=-2

y''(0)=-\dfrac{11}2=C_2+2C_3-6+\dfrac{25}2\implies C_2+2C_3=-12

\implies C_1=10,C_2=-10,C_3=8

and so

\boxed{y(x)=10-10e^x+8xe^x+2x-12x^2e^x+\dfrac12e^{5x}}

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Answer:

<u>A. The equation describing the relationship between speed and time is:</u>

<u>s = 32t</u>

<u>Ordered pairs of the form (speed, time) are:</u>

<u>(32, 1)</u>

<u>(64, 2)</u>

<u>(96, 3)</u>

<u>(128, 4)</u>

<u>B. </u>

<u>Using s = 32t, we have:</u>

<u>s = 32 * 5 = 160 feet per second</u>

Step-by-step explanation:

1. Let's review the information provided to us to answer the question correctly:

Height of the cliff = 500 feet

Speed after one second = 32 feet per second

Speed after four seconds = 128 feet per second

2. Assume the relationship between time, t, and speed, s, is linear and write an equation describing the relationship. Use ordered pairs of the form (speed, time)

The equation describing the relationship between speed and time is:

s = 32t

Ordered pairs of the form (speed, time) are:

(32, 1)

(64, 2)

(96, 3)

(128, 4)

3.  Use the equation to determine the speed of the rock 5 seconds after is dropped.

s = 32t

s = 32 * 5 = 160 feet per second

Note: Same answer to question 14436233, answered by me today.

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Answer:

y = 2x + 1

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first find slops

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5 0
3 years ago
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Answer:

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Step-by-step explanation:

We have been given that college students average 8.6 hours of sleep per night with a standard deviation of 35 minutes. We are asked to find the probability of college students that sleep for more than 9.6 hours.

We will use z-score formula to solve our given problem.

z=\frac{x-\mu}{\sigma}

z = z-score,

x = Random sample score,

\mu = Mean,

\sigma = Standard deviation.

Before substituting our given values in z-score formula, we need to convert 35 minutes to hours.

35\text{ min}=0.58\text{ Hour}

z=\frac{9.6-8.6}{0.58}

z=\frac{1}{0.58}

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