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1_ (10 Marks) Obtain the general solution of the system below by the method of DIAGONALIZATION.2t2 X +X =e2t...

Question

1_ (10 Marks) Obtain the general solution of the system below by the method of DIAGONALIZATION.2t2 X +X =e2t

1_ (10 Marks) Obtain the general solution of the system below by the method of DIAGONALIZATION. 2t 2 X + X = e2t



Answers

Use the operator method described in this section to find the general solution of each of the following linear systems. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+\frac{d y}{d t}=e^{2 t} \\ &\frac{d x}{d t}+\frac{d y}{d t}-x-y=0 \end{aligned} $$

In the problem we have been given dx upon DT minus the way upon DT minus two X plus four way that is equal to t. And the other is the X. Up on duty plus. They weigh upon in DT minus x minus Y. Which is equal to one. So this becomes D -2 X minus the minus four Y. Which is equal to t 1st equation. And then D -1 X plus d minus one way. That is equal to one. This is the second equation. So further with missile went find for ex therefore the first question is multiplied with the minus wanna And then we have two plus it with the minus four into second equation. So we have d minus one And today -2 x- and the -1 and two d -4. Y. That is equal to D -1. D rather this is like this now we have further yeah D -1 into the -4 X plus The -1 into the -4. Y. She is equal to D -4 into one. I have to subtract this story and this part soft reading, we omit this one and we have finally They squired minus day -2 day plus two plus The Squire -1 -40 plus four X. And that gives us 1 -1 blast minus four. So we have this equals two. The Squire minus three D plus two Plus The Square -5 D. Plus four eggs. That is equal to 1 -5 -4. So we have so we have this and now the general solution to the differential equation becomes X. Equal 7 8. The party plus see to it about treaty -13 upon 18 -1 upon 60. Yeah so further we have to find for the value of why and now mhm. We have they Tuition's one and 2. So after doing this we have these equals two The -2 plus the -1. X. Plus minus D. Plus four plus the minus one. Y. That gives us T Plus one. These kinds of this part. And we have to the minus three X. Plus three way that is equal to T plus one. So three Y is equal to D plus one minus two d -3 X. Further the value of say Y is equal to T plus one minus two into dx so dx 78 party plus three C. To the power treaty -1 About six. So we have Y equals one upon 3 into T plus one minus two and two. So we have Y equals one upon three into T plus one minus two into someone eat party plus three C. To eat part three D minus one upon six plus three into 78 party plus C. Two and part three D minus 13 upon it in minus t upon six. So we have this as the answer to the problem

In the problem we have D minus one X. Plus de plus five White. That equals the squad. Nowadays. Day -2 X. Plus to the plus four Y. That equals two two d. Plus one. So we'll be multiply That is two d. plus food. With the first equation. Then subtract them implying D. Plus five. With second equation. Therefore we have to d. plus four Into the -1 x. Plus two d. Plus four. Today Plus five Y. That equals two two d. Plus four dear squad. Now we have deep plus five D minus two X. Plus 20 plus four day plus five Y. That equals two D. Plus five To T. Plus one. Now here. Mhm. You have to subtract so this part Kansas each other and now we have to the Squire -2 d. plus 40 minus four minus the esquire plus five d -2 D minus 10 X. That equals two 40 plus 40 square -2- Danity -5. Mhm. So further it is equal to the a squared plus two d -3 D -4 plus 10 X. That equals to 40 plus 40 square minus two minus Dante minus five. Mm Father it is the Squire minus D. Plus six X. That equals to 40 Squire -60 -7. Mhm. Or we have the X. That is the general solution to the different cell equation becomes X. That is equal to eat power T. Upon to And to see one. Sign. Route 23 upon to t plus stay too cause Route 23 upon to T minus for do you wanna bon 27 -14 upon 18 d. Plus two upon 3 d. squad. Mhm. So further we have to reply to with the first equation and then subtract the second equation. Therefore we have julie -2- Deep Glass two X plus two D. Plus 10 -2 d -4. Why? Which is equal to 2? T sq -2 T -1. Or we have D. X. Plus six Y. That is equal to To T. Esquire -2 T -1. So we have six white and there's equals two to T Squire minus two t -1 of- & DS. or why equal one upon 6 into to D squared minus two t minus one and minus and dx. Further we have to simplify four DX. And then we will put this value in this aggression the question so that we have to obtained finally the value of what you mm so finally we have Y equals one. About 16 to 2 day square -2 T -1 -DX. So this is the value of DX. And we have to put this well here so after putting it we obtained this as mm minus takes so d exist half it party upon to into seven sign 123 upon to T plus C. Two cars 123 upon to T plus eight. Party upon to into route 123 upon to seven cost 223 upon to t minus 23. Route under upon to see to sign 123 by two T. So this is the final answer to the problem.

In the problem we have been given To the -1. X plus the minus one Y. That equals one. This is first equation. 2nd 1 is deep plus two X plus d minus one Y. That equals t. This is our second equation Now 1 -2 becomes mhm. To the -1 -3 -2. X plus the minus one minus t plus one Y. That equals 1- T. So we have Day -3 X. That equals 1- T. Our X equals C. Power treaty Plus D. Upon 3 -2. Up online. Now we obtained this value of X. So this is X. S. D. General solution to the differential equation. Now we have to obtain why So to obtain why we have to do as directed that is D Plus two is multiplied with the first equation and minus To the -1 is complied with the second equation. So for that we have mhm D Plus two and 2. Today -1 X plus de plus two and two d minus one. Y. Which is equal to D Plus two and 1. And here it is two d -1 and two D plus two X plus To the -1 in two d minus one that equals two, two d -1 and duty. Now we subtract these equations. So in this cancels each other. No we have here the esquire plus Today -D -2 minus today Squire minus d minus today. Last one. Why? Which is equal to two minus two DT minus t. So that's become the Squire plus the minus two minus two. The Squire Plus three D -1 Y. That is equal to 2 -2 plus city. And this cancel this part. So further it is minus the esquire plus 40 -3 Y. Which is equal to T odd. We have They Squire -40 Plus three Y. Which is equal to T now. Further so here mhm. We have the Y equals Okay one at the party plus. Okay to help our treaty minus D. A boundary -4 Upon night. Now we have to find the value of kevin and ghetto in terms of C. Because here we have the value of C. I mean the constant see here so we have to find kevin kevin and ghetto in terms of seven R. C. Two. So first of all we have to put the values of X and Y. In equation fast. So this is our first question. That is studio x minus x plus D. White minus Y. That equals one. So we will write the values of X. And Y. In this 1st situation. Therefore this becomes mm Mhm two decks minus six plus D. White- Why that equals one. Now it becomes too into D. Access three C. E. The Power Treaty Plus one Abound 3 minus say is the power treaty minus the abound three Plus to about nine plus given into the party plus three K. Two. It part treaty -1. up on three minus. Given you the poverty minus K. two Power 3 d. Plus the upon three plus 4.9 that equals one. Now after solving we obtained This has six ce power treaty Plus one. A boundary minus C. Is part treaty Plus two upon 9 Plus two K. to its power three D. Plus four Upon mine that equals one. Now for that We have this as five ce power treaty plus two K. to eat power treaty plus 2.3 plus £1.03 that equals one. Or we have five c. Plus to kato. It powered treaty that equals zero. Now we have five. See that because minus to get to odd Okay two can be written as -5 c. two or minus five C upon do. Therefore we have the key to informs of see which is minus five C upon two. And we keep kevin arbitrary as see two. So finally we have the value of why that equals given is C. Two your power T plus K. Two. So Kay to is minus five C. Upon to Eat about three T -D upon 3 -4 upon nine. So finally we have this as the answer to the problem

In the problem we have the squad X minus dy that equals t plus one. This is first equation And we have second question as D -3 X Plus de plus one. Why? That is equal to To T -1. This is our second equation. Now when we went by the first equation with D plus one and add by applying deal with second equation. Therefore we have this as a D plus one and today Squire x -D plus one and two D. Y. That is equal to D plus one and two T plus one. And here it is the Squire minus three D. X plus D. Plus one D. Y. That is equal to day off two D. Plus. Sorry this is minus. So yeah it is minus one. No as we add these two equations, these cancers each other and here we have de que plus D esquire plus the Squire minus three d. X. That has given us one plus t plus one plus two. No We have this equals two de que plus two Day Squire -3 D. X. That is equal to four plus d. Now here it is D into the squared plus to the -3 x. That is equal to four plus day. Or here we have d. in two D -1 in two D. Plus three X. That has given us four plus t. Or we have x equal to seven plus C. Two. Eat party plus C. three. Eat bar minus three D -4.3. So this is our X. And further we have the X Equal to seven Plus c. two. Eat party plus C. Three. Eat Power minus treaty -4.3. Now we have to find the value of why? So we'll add 1st and 2nd equation. Therefore this gives us D squared plus d minus three X. That plus dy plus y minus dy Which is equal to D. Plus one Plus 2 T -1. These kinds of this part this cancer of this part And further we have the Squire plus D -3 X. Plus. Why? Which is equal to three D. Now why he becomes three t minus D. Square x minus dx plus three X. So we have y equal to three D minus C. To eat party minus nine C. Three. Eat bar minus treaty minus C. To eat party plus three C. +38 par minus treaty Plus three C 1 Plus three C. to eat party Plus three c. 3. Eat bar -3 D minus four. So further why becomes three C. One plus C. To eat poverty minus three C. Three Eat bar minus treaty plus three d -4. So this is the answer to the problem


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