Open study

is now brainly

With Brainly you can:

  • Get homework help from millions of students and moderators
  • Learn how to solve problems with step-by-step explanations
  • Share your knowledge and earn points by helping other students
  • Learn anywhere, anytime with the Brainly app!

A community for students.

Differential Equations-> Laplace-> Piecewise Defined functions. Solve the initial value problem.

Mathematics
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Join Brainly to access

this expert answer

SIGN UP FOR FREE
1 Attachment
L{y'+3y} = (s - 3)Y(s) - 1.\[L \left\{ f(x) \right\} = \int\limits_{0}^{\pi}e ^{-s x}\sin x dx\]\[= (e^{-\pi x} - 1)/(s^{2} + 1),\]so \[(s - 3)Y(s) - 1 = (e^{-\pi s} - 1)/(s^{2} + 1),\]giving us\[Y(s) = 1/(s + 3) + \[(e^{-\pi s} - 1)/[(s^{2} + 1)(s + 3)]\].\]No problem finding \[L^{-1}\left\{ 1/(s + 3 \right\}.\]To find\[L^{-1}\left\{ (e^{-\pi s} - 1)/[(s^{2} + 1)(s + 3) \right\}, \]we can use the convolution theorem. We know that\[L^{-1}\left\{ (e^{-\pi s} - 1)/(s^{2} + 1)\right\} = f(x),\]so the inverse laplace transform works out to \[f(x) * e^{-3x} = \int\limits_{0}^{x}f(u) e^{-3(x - u)}du.\]
\[= e^{-3x}\int\limits\limits_{0}^{x}e^{3w} \sin w dw\]\[= (e^{-3x} + 3 \sin x - \cos x)/10.\]The solution, then, is\[y(x) = (e^{-3x} + 3 \sin x - \cos x)/10\]

Not the answer you are looking for?

Search for more explanations.

Ask your own question

Other answers:

for \[x \in [0, \pi), \]\[(e^{-3 \pi} + 1)/10, x \in [\pi, \infty).\]
ewwww, my solution was wayyyy off.....we haven't learned convulution theorem yet....think we're covering that next. I'll post my solution in a minute....I tried working it out along with my professor's video.
My solution is 3 pages long....somebody please take a look at it and give me some feedback. See attached.
Page 1
1 Attachment
Page 2
1 Attachment
Page 3 (sorry, it was giving me trouble when trying to upload all 3 at one time.)
1 Attachment

Not the answer you are looking for?

Search for more explanations.

Ask your own question