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Bond [772]
3 years ago
5

Find the general solution of the following ODE: y' + 1/t y = 3 cos(2t), t > 0.

Mathematics
1 answer:
Margarita [4]3 years ago
3 0

Answer:

y = 3sin2t/2 - 3cos2t/4t + C/t

Step-by-step explanation:

The differential equation y' + 1/t y = 3 cos(2t) is a first order differential equation in the form y'+p(t)y = q(t) with integrating factor I = e^∫p(t)dt

Comparing the standard form with the given differential equation.

p(t) = 1/t and q(t) = 3cos(2t)

I = e^∫1/tdt

I = e^ln(t)

I = t

The general solution for first a first order DE is expressed as;

y×I = ∫q(t)Idt + C where I is the integrating factor and C is the constant of integration.

yt = ∫t(3cos2t)dt

yt = 3∫t(cos2t)dt ...... 1

Integrating ∫t(cos2t)dt using integration by part.

Let u = t, dv = cos2tdt

du/dt = 1; du = dt

v = ∫(cos2t)dt

v = sin2t/2

∫t(cos2t)dt = t(sin2t/2) + ∫(sin2t)/2dt

= tsin2t/2 - cos2t/4 ..... 2

Substituting equation 2 into 1

yt = 3(tsin2t/2 - cos2t/4) + C

Divide through by t

y = 3sin2t/2 - 3cos2t/4t + C/t

Hence the general solution to the ODE is y = 3sin2t/2 - 3cos2t/4t + C/t

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Wendy goes to the gym every 4 days and swims every 6 days. If she went to the gym and swam today, in how many days will she do b
Mice21 [21]

In 12 days, Wendy will do the both again.

Step-by-step explanation:

Given,

Wendy goes to gym every 4 days

Wendy swims every 6 days

We will find the least common multiple of both the numbers in order to determine the number of days in which she will do both.

4 = 4,8,12,16,20,....

6 = 6,12,18,24,.....

Therefore;

The least common multiple of 4 and 6 is 12.

Thus,

In 12 days, Wendy will do the both again.

Keywords: multiples, LCM

Learn more about multiples at:

  • brainly.com/question/10196212
  • brainly.com/question/1021953

#LearnwithBrainly

6 0
3 years ago
Can I get some help please? <br><br><br> Thanks!
quester [9]
An ellipse (oval shape) is expressed by the following equation:

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1 where h is the x coordinate of the center and k is the y coordinate of the center. Furthermore, a is the horizontal distance from the center, and b is the vertical distance from the center. Lastly, c is the distance from the center to one of the foci (they are spaced apart equally).

We can find the foci by using a^2 - b^2 = c^2

36 - 11 = c^2

c =  \sqrt{25}  = 5

Since the k value in this case is 0, the y value of both foci are 0. Also, since h and k are both 0, we know the center of the ellipse is at the origin.

So the foci are (-5, 0) and (5, 0)

Hope this helps :)

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3 years ago
Its pretty easy if you are good at math! Please help!
aleksley [76]

Step-by-step explanation:

Explanation:

The trick is to know about the basic idea of sequences and series and also knowing how i cycles.

The powers of i will result in either: i, −1, −i, or 1.

We can regroup i+i2+i3+⋯+i258+i259 into these categories.

We know that i=i5=i9 and so on. The same goes for the other powers of i.

So:

i+i2+i3+⋯+i258+i259

=(i+i5+⋯+i257)+(i2+i6+⋯+i258)+(i3+i7+⋯+i259)+(i4+i8+⋯+i256)

We know that within each of these groups, every term is the same, so we are just counting how much of these are repeating.

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From here on out, it's pretty simple. You just evaluate the expression:

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The answer is on the paper

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