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(12 pts) Using the Laplacc Transform, solve the IVP:fo 10 < t<t %" + y sin (2t) T<t <0Oy(o) 0...

Question

(12 pts) Using the Laplacc Transform, solve the IVP:fo 10 < t<t %" + y sin (2t) T<t <0Oy(o) 0

(12 pts) Using the Laplacc Transform, solve the IVP: fo 10 < t<t %" + y sin (2t) T<t <0O y(o) 0



Answers

Use the Laplace transform to solve the given initial-value problem. $y^{\prime \prime}-y^{\prime}-2 y=10 \cos t, \quad y(0)=0, \quad y^{\prime}(0)=-1$.

In the problem we have to what level does plus? Why does minus Y? Which is equal to sign treaty? Why F0 is equal to 0? Why does zero is equal to zero now? Had. So here we obtain the lap lots of why that is equal to three upon as a Squire plus nine in the to s minus one into S plus one. Now as this is a partial fraction so this can be solved here like this. So this is represented as three upon It is squared plus nine into two hours -1 into S plus one equals two. Airport S plus one plus They upon to S -1 -1 plus CS plus D. Upon esquire plus nine. No. Using hit and trial method we have solved obtained the result. That is a equal to -1 upon 10. Be equal to eight upon 37 C equals two -43.370 Day equals two 23 upon 3 70. Now, further this is served as lipless Universe -1 upon then into s plus one plus eight upon 37 into one. Upon Going to S -1 upon to plus -43. Bomb 3 70. Yes plus 23 upon 3 70. You want to buy? It's a squared plus nine. No this become -1 upon 10. With the power minus T Plus four upon 37. Heat Power T upon to -43. S upon 3 70 into esquire plus three square Plus 23 pond 3 70 in two it's a squared plus three squared. Now further this equals two -1 upon 10. It bar -1 plus four upon 37. A party upon to minus 43 upon 3 70 of course 30 Plus 23 upon 3 70. Sign 30. So this is the answer.

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.

So mr off this problem by changing the way I write the Y. Prime. So why prominent writer instead of Dy over DT? And that equal slutty sign T squared plus one. And so now I'm gonna multiply both sides by D. T. So why D. Y. Equals the T. Sign T squared plus one T. T. And I'm gonna integrate both sides here. So the integration of why is Y squared divided by two. And for the right hand side I'm going to use a U. Substitution. So I'm gonna use U. Equals T square plus one D. You will equal to two T. So in order to satisfy um this U substitution, I would have to multiply it by a half. So I'm going to have one half to t. Sign T squared plus one T. T. And now I can fact or rather substitute this. You and eu terms in. So I'll have white square divide by two equals to one half. If you use the substitution if you get a sign of you, do you? And the integration of sign of you is negative. Co sign If you we're going to have one half negative one half. Co sign a few plus C. And if we plug back in U. Equals the T square plus one. We'll get that by square divided by two equals two. Negative one half. Co sign of T squared plus one plus C. And now let's multiply both sides by two. So why squared equals two negative co sign T squared plus one plus C. And let's take the square root. So we'll have that Y equals two. The square root plus or minus negative. Co sign T squared plus one pussy. And that's your final answer.

In the problem they have Why double glass minus four Y. That is equal to three costea. Y. Of zero is equal to zero. And why does zero is called 20 So this is written as esquire lap lots of y. Now here this is the partial fraction form. So we have to obtain values of A B and C. So A is regionals. This is the value of A, B, C and D. So the value of A becomes 3.10. 3.10. So this is obtained by hidden trial method. Now we will find the hapless inverse. So lap listen words become 3.10. No I did. That is true upon turn into A. S plus two Plus three upon 10 And the S -2 plus -3.5 is upon As a Squire plus one. So this is region as 3.10, you'd power minus duty Plus three upon 10 years. The power to T -3 upon five Costea. So this is our Y f T. And this is the answer to the problem.


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