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lys-0071 [83]
3 years ago
9

Find the solution of the given initial value problem. ty' + 2y = sin t, y π 2 = 9, t > 0 y(t) =

Mathematics
1 answer:
Helen [10]3 years ago
6 0

For the ODE

ty'+2y=\sin t

multiply both sides by <em>t</em> so that the left side can be condensed into the derivative of a product:

t^2y'+2ty=t\sin t

\implies(t^2y)'=t\sin t

Integrate both sides with respect to <em>t</em> :

t^2y=\displaystyle\int t\sin t\,\mathrm dt=\sin t-t\cos t+C

Divide both sides by t^2 to solve for <em>y</em> :

y(t)=\dfrac{\sin t}{t^2}-\dfrac{\cos t}t+\dfrac C{t^2}

Now use the initial condition to solve for <em>C</em> :

y\left(\dfrac\pi2\right)=9\implies9=\dfrac{\sin\frac\pi2}{\frac{\pi^2}4}-\dfrac{\cos\frac\pi2}{\frac\pi2}+\dfrac C{\frac{\pi^2}4}

\implies9=\dfrac4{\pi^2}(1+C)

\implies C=\dfrac{9\pi^2}4-1

So the particular solution to the IVP is

y(t)=\dfrac{\sin t}{t^2}-\dfrac{\cos t}t+\dfrac{\frac{9\pi^2}4-1}{t^2}

or

y(t)=\dfrac{4\sin t-4t\cos t+9\pi^2-4}{4t^2}

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Help complete the table for the equation y= x - 2. Then use the table to graph the equation.
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Answer:

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Step-by-step explanation:

Using the equation y = x -2, substitute the values for x into the equation and solve for y.

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Graph the line using a point and a slope. Write the equation of each line. A line that passes through the point (0, –3) and para
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5 0
3 years ago
An object is dropped off a building that us 144 feet tall. After how many seconds does the object hit the ground? (s= 16t^2)
denis23 [38]
Given 
s=16t^2 
where
s=distance in feet travelled (downwards) since airborne with zero vertical velocity and zero air-resistance
t=time in seconds after release

Here we're given
s=144 feet
=>
s=144=16t^2 
=> 
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so
t=3
Ans. after 3 seconds, the object hits the ground 144 ft. below.
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Answer:

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Step-by-step explanation:

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