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Use Laplace transforms to solve the differential equationd20 do 16 640 = sin (6t) dt dtgiven that 0 and its derivative are zero at t= 0....

Question

Use Laplace transforms to solve the differential equationd20 do 16 640 = sin (6t) dt dtgiven that 0 and its derivative are zero at t= 0.

Use Laplace transforms to solve the differential equation d20 do 16 640 = sin (6t) dt dt given that 0 and its derivative are zero at t= 0.



Answers

Use the Laplace transform to solve the given initial-value problem. $y^{\prime \prime}-y=8 \sin t-6 \cos t, \quad y(0)=2, \quad y^{\prime}(0)=-1$.

Do you think? The Function producer. He's from thes differential play. Why? Double prime? Why you two times signed e e seeing why zero cools five here. Alas, what? So he's individual equation will be rearranged to be you arrange to be wise for in the posh informed Why asked hands It's minutes one day Going to as minus one was eight million. Six best Why? Why us will be equal? Teoh two has cute My has Okay as one His plus one in s where rich, deceitful to equal Teoh miss us His one plus one This finest one. Okay, square Isis. All right, Fielding Thio Thio has four time

Our first step here is going to be seeing little plus transform their differential equation. We do. So we're going to get the flash transform of the second derivative minus two tons of glass, transforming the first derivative minus eight time. Some Russians worldwide being angles in the class, transform of just five. And we can use our rules or the plus transforms of conservatives to simplify this. To get that this is s square turns capital Y minus s friends wave zero minus y private zero, which is their first term minus two times essence capital. Why minus y zero minus eight times Cavalli is equal to applause. Transform of five. But you can pull find outside together a posh transform of wanting which had nothing todo zero, which is five times zero factorial which is one over s. Now we're gonna factor out a capital. Why, When we do so we're gonna getting s squared minus us minus eight turns capital. Why minus s why prime of zero is zero plus two is equal to five over s We're gonna add in essence, attract too. And then divide by the factor in front of capital. Why? To get the capital. Why is find over s plus s mines too. All about of my ass squared minus two s minus eight. And now we can find a common denominator of ass in the numerator and then bring it into this nominator together this five, where s squared? Minus two s plus five in the numerator, all divided by s times this denominator which we can factor to be s minus four times as plus two. And now we're gonna shut it right this as three separate factions, a over s must be over s minus four. So I see Over s plus two and we're gonna multiply through by the denominator here to get that s squared minus two s plus five is equal to a times s minus four times x plus to you, plus being times s plus two kinds. That's plus seeing turns s times X minus four. And now we're gonna write down blue. So special cases for s So if past is equal to zero on the left will have zero squared minus zero plus five, just five being equal to a times negative floor times two plus zero plus zero. Therefore, a is negative. Five over eight, and then we can set s equal to four on the left hand side and we're gonna get 16 minus two tens for his eight plus five is equal to 13 is equal to any time. Zero plus being times four plus two is six times for 10 There will be is eagled on 13 November 20th floor. And then finally, we said s equals negative, too. Wearing it, *** two squared is four minus two times negative. Two is a positive floor. Plus five is equal to 13 which is equal to a times zero plus B time. Zero plus seven times negative, too. Times negative six and therefore see is equal to you. 13 over 12 in therefore our capital. Why is equal to you? 13/25 times won't ever asked. Minus four plugging in these coefficients. Plus 13/12 turns one over s plus two minus 5/8 terms. One address. Now we're gonna take the inverse of loss transform of both sides to get that R y A t. Because Eagle to 13/25 times the inverse laplace transform of one of the s minus four plus 13/12 terms. Endorse applause, Transform of one of her ass. Suppose to minus 5/8 times and response transform of one of us. And now we can use our other rules up care for each of the s and T raised to the power to get this is nothing other than you need to the floor. 14. And this is need to the negative to G. So it is for in this case and making it to in this case. And this is nothing other and just one or immigration zeroth power, which is just one. It's therefore Y t is anyone 13/25 times each of the 14 plus 13/12 times and even negative fut minus five of rate.

In the problem we have. Do Y. Double bass plus eight Y. That it was three sign to T. Y. Of zero is equal to zero. And why does zero is equal to zero? So it is written as two and 2. It's a square la place of white minus S. Y. Of zero minus vilas zero plus eight. Into a place of white. That equals to three into two upon as a squared plus four. Softer arranging. We have this as to into esquire la plus off. Why? Sorry, This is return us. Mhm. to esquire plus eight Lab plus of why? That equals to six upon. It's a square plus four. So this is like a place of why you become three upon is a squared plus four into esquire plus four. So the city lipless. Now you have to find the lab plus universe. Therefore, further we have. Mhm. Y. Ft. That becomes lifeless and worse of three upon S squared plus four To the power of two. Now, yeah, it's become the pleasant worse three and 2 one up on 16 in two to into eight upon A squared plus two Squire to the power of two. This is equal to 3.16 into Signed to T -2 T. Cost to T. This equals three upon 16 sign duty And three upon 8 d. Cost duty. Therefore, Y F T equals this and this is our answer.

You trained soft with the potential pleasure. I don't Crime thing is equal. Six teams co saying team using your why sear girl and prime here past transform of INGE s cleared. Why, yes. Poor is has not why has here four divided by one s queer plus one was quite this one. It's really simple but part of Clinton to form up. Right? You. I want what a wonderful By s ways you have 11 three as That's e e t. Thank you. Three she


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