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hjlf
3 years ago
8

2 points) Sometimes a change of variable can be used to convert a differential equation y′=f(t,y) into a separable equation. One

common change of variable technique is as follows. Consider a differential equation of the form y′=f(αt+βy+γ), where α,β, and γ are constants. Use the change of variable z=αt+βy+γ to rewrite the differential equation as a separable equation of the form z′=g(z). Solve the initial value problem y′=(t+y)2−1, y(3)=4.
Mathematics
1 answer:
Stells [14]3 years ago
3 0

y'=(t+y)^2-1

Substitute u=t+y, so that u'=y', and

u'=u^2-1

which is separable as

\dfrac{u'}{u^2-1}=1

Integrate both sides with respect to t. For the integral on the left, first split into partial fractions:

\dfrac{u'}2\left(\frac1{u-1}-\frac1{u+1}\right)=1

\displaystyle\int\frac{u'}2\left(\frac1{u-1}-\frac1{u+1}\right)\,\mathrm dt=\int\mathrm dt

\dfrac12(\ln|u-1|-\ln|u+1|)=t+C

Solve for u:

\dfrac12\ln\left|\dfrac{u-1}{u+1}\right|=t+C

\ln\left|1-\dfrac2{u+1}\right|=2t+C

1-\dfrac2{u+1}=e^{2t+C}=Ce^{2t}

\dfrac2{u+1}=1-Ce^{2t}

\dfrac{u+1}2=\dfrac1{1-Ce^{2t}}

u=\dfrac2{1-Ce^{2t}}-1

Replace u and solve for y:

t+y=\dfrac2{1-Ce^{2t}}-1

y=\dfrac2{1-Ce^{2t}}-1-t

Now use the given initial condition to solve for C:

y(3)=4\implies4=\dfrac2{1-Ce^6}-1-3\implies C=\dfrac3{4e^6}

so that the particular solution is

y=\dfrac2{1-\frac34e^{2t-6}}-1-t=\boxed{\dfrac8{4-3e^{2t-6}}-1-t}

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Answer:

0.40 foot

Step-by-step explanation:

In the first case, a 50 - foot building casts a- foot shadow (1-foot). Let the angle of elevation of the sun from the shadow be represented by θ.

Then:

Tan θ = \frac{opposite}{adjacent}

Tan θ = \frac{50}{1}

Tan θ = 50

⇒ θ = Tan^{-1} 50

      = 88.8542

      = 88.85^{o}

The angle of elevation is approximately 88.85^{o}.

For a 20-foot pole,

Tan θ = \frac{opposite}{adjacent}

Tan 88.85^{o} = \frac{20}{x}

x = \frac{20}{Tan 88.85^{o} }

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The length of the shadow of the pole is 0.40 foot.

4 0
3 years ago
Express it in scientific notation.
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Multiplying the values given above we obtain a value of 0.01386. In scientific notation, 1.386 x 10^-2. <span>The </span>significant figures<span> of a number are </span>digits<span> that carry meaning contributing to its measurement resolution. From the multiplicand and the multiplier, the least number of significant significant figure would be one. Therefore, we write as 

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An art museum had 275,760 visitors last year. The museum was open every day of the year except for 5 holidays. How many visitors
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Step-by-step explanation:

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Need help l don’t understand could someone help me?
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Step-by-step explanation:

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Mae has a number cube with 6 sides that are numbered 1 through 6. What is the probability that Mae will roll an odd number?
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Answer:

C) P(E)  = \frac{3}{6}<u></u>

The probability that Mae will roll an odd number

<u></u>P(E)  = \frac{3}{6}<u></u>

Step-by-step explanation:

<u><em>Step(i):-</em></u>

<em>Given that Mae has a number cube with 6 sides that are numbered 1 through 6.</em>

<em>n(S) = { 1,2,3,4,5,6,} = 6</em>

<em>Let 'E ' be the event of odd numbers</em>

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