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Solve the initial value Problem usingvariation of Parametersy" y"' = 2t, Azach FileYo) = 0, Y(0) =0, Y'(o) = -2.BroiseFilesBrowze Conten: Collec...

Question

Solve the initial value Problem usingvariation of Parametersy" y"' = 2t, Azach FileYo) = 0, Y(0) =0, Y'(o) = -2.BroiseFilesBrowze Conten: Collection

Solve the initial value Problem usingvariation of Parameters y" y"' = 2t, Azach File Yo) = 0, Y(0) =0, Y'(o) = -2. Broise Files Browze Conten: Collection



Answers

Solve the given initial-value problem. $$y^{\prime \prime}-4 y=2 t e^{t}, \quad y(0)=0, \quad y^{\prime}(0)=0$$.

This question asked us to find the solution of the differential equation that satisfies the given additional condition. We know we have d'you over G is two teeth plus sequence where team divide by to you. Now let's get all the you terms on left inside to you D you and all the ti terms on the right hand side because it will make it significantly easier to integrate. Take the integral of both sides. Tiu to you becomes you squared because we increase the exploited by one and we divide by the new exponents. Marco, fishing just becomes one or two over too. On the right hand side, we have cheese squared plus tan of team plus c Remember, the integral seeking scored of tea is 10 of tea. Now that we have this, we know we're gonna be substituting. And if you have zero is negative five than t zero and use negative five substitute in. Remember, we're solving for C. We got C is 25. Lastly, plug back in to our equation that we determined once we integrated the sea is 25 instead of plus C, we now have plus 25 then take the squirt of both sides. Because we want this in terms of singular. You not. You squared and we end up with our solution. So this is all under the square root now?

Hello, everyone. Today we are going to sort problem Number 34 this is we have to find it in this leper stand for that is you have an initial value problem. The given our by double s my last life because you were not be. And while I was little it goes to And why would I sell zero? It goes little. So taking leftist transall or boots is as a square by all of us. Quietness says, while I was little my less by myself syrup my enough well I office because that is you learned after that is fair play. It's OK, shift in India. Very good. You didn't bar minuses, but yes, Why are zero is too. Where myself zero is cereal so combating We can try to test my office equals he Mr Minuses. But as in the as a wrestler in the s minus 11 Bless minus two s or my coastal right and second chance place Onder there be as Pressler in the s my uniform. So by playing partner fashion with 30 for this and it's for you can write business and by us for us B by us. Let's learn plus c by as Max on this kind Return us. Do you buy? Yes. Bressler left you by F minus. Learn so it can be forward Breath blinded by as a square minus one. Some stewed as inconsiderable. That is usually we get the nest. My enough like big us one by his Bressler that you can be found ordered by one by yes, Indo as my substitute as incomes minor form. It's a big getting us like by foot. Similarly See is a good one by s Indo. If I flirt Substitute Asik Whistler were bigger tinnitus one by similarly the because us there by that's minus lock Some stood as because my but because to less developed by yes pleasure. Some stood a single one which is in Cordoba. Work, Phil, my office and return us my enough one. But as you know, the border by an SS less one by you Did a par minus sense there, right by as this long plus one by do you does a par minus is they aren't but s minus one Bless learn by s press on press learn by AZ minus up. Do you think you can be sure? Followed by applying so good shifting Jenna if because of Futebol bonuses they were very impressed by the metaphor. And my dear brand by you're enough big So our biases the you are not take plus one bite contributors to minus T Do you think best buy do you, my Nesler Inter, You are enough tea. I'm not late plus one by two certainly in the u necessity. But he added Will be day My Nesler, You are Loveday Freeman and things is minister minus T plus the list that if by often that wasn't awful question Thank you.

Okay, so here we have a non homogeneous differential equation on. Uh, we want to solve this by and determine coefficient matters. So first of all, we want to find the compliment, the solution. And for that, uh, right, the ox Hillary equation for the homogeneous system. So we can right here. A square minus four r is equal to zero. Now we can factor out, are so here are modified by our Marla's four is equal to zero. Therefore, it is equal to zero, and all is equal to four. Right. So here we have a two. Really good. So far, this complemented solution is in the form off. See? One. It is 20 x plus C toe. It is +24 X Here it is. 20 Ecstatic called one. So we can right here only c one plus c toe it is to forex. Okay, Now, on the right side of the d, we have a It is two x multiplied by cause affects. So let's assume that our particular solution is in the form of why is it called tau zero? Is it? It is two x multiplied by sine X plus a one years to eggs multiplied by caused X. Now take a daily with you. So you're why this is equal toe zero multiplied by eaters to X in X plus it is two X Cossacks less a one multiply by. It is two X Cossacks minus it is to break cynics. Now take a double the radio. So you're Y double s. That is equal door to either. Is two weeks multiplied by 80 Cost affects minus a one. Sign off X right now Substitute the value off. Wider, bullish And why days in this different cell equation. So our differential equation is why the about this minus why this is equal toe. It is two X Cossacks. Right now your wider bowl dish that is to eat is two x multiply by zero cause off X minus a one sign off X and now substitute the value of wireless And now simplify this equation. So here we have a negative to a zero. It is two X Cossacks minus for a zero. It is two x in X plus to a one. It is two x in X minus For a one it is to Xcor sex and on the right side is as it is right now again simplify this so we can Right Here it is two x Synnex multiplied by negative for zero plus to a one And here it is to Exco Sex multiplied by negative two ways you know, negative 41 And on the right side that is it is to Exco Sex plus zero It is two x sine x and that's equal to zero right? So we can write like this Now compare the coefficient both side So first to follow we need to compare the coefficient off It is two x in X here on the right side The coefficient off arrest Rexene X zero right And now compare the coefficient off others to Exco Sex now on the right side The coefficient off it is to Exco Sex is one right now by solving both the situation So we have a zero is equal to negative one by 10 and everyone is equal toe negative one by five Now substitute toe both this constant in our particular solution So our particular solution wipe that is equal to negative one by 10 years to make cynics minus one by five years to X core sex. Right now our general solution is given by y c plus y p right here, vices, our complemented solutions. Our complemented solution was see when it is to forex plus C toe minus one by 10 It is two x in X minus one by five It is two x core sex. Okay, now here Our initial condition is why off zero is equal to one, which means substitute X is equal to zero. And why is it called one in this equation? So your wife is equal to one. Here it is. 20 That's zero. So here we have an only C one plus C toe minus one, divided by 10. Here, Sign zero is equal to zero and zero multiplied by one by 10. That is equal to zero. So we can right here. Zero only, minus one by five. Right, So here are C one is equal to negative. See? Two black, 60 added by five. Right. Okay. Now substitute the value off. See one in this equation. So here. Why is it called toe negative? C two plus six divided by five. It is 24 x class C to minus one by 10 It is two x in X minus one by five It is two X Cossacks now take a daily ready off dysfunction. So here, why This is equal to negative forces to us to forex minus three by 10 years to Exco Sex plus one by 10 it has two X in X plus 24 by five years to four x Okay, here. Second initial condition is why there's a zero is they called to write. So here substitute X is equal to zero and why this is a call to to so here to is equal to negative 42 minus three by 10 plus zero plus 24 by five Right, So here, see to that is equal to find my 80 right now Substitute the value off Cito in this equation. So general solution is why is it called toe 23 by 10 or your 23 by 14 or 10. It is 24 x bless The R C two is five by eight minus one by 10 It is two x in X minus one by five years to x Cossacks, That is to your final answer. Thank you

Hello. So the question here belongs to the first order differential equations. So given that why climb plus two Y. Is equal to team two. To the power minus duty with initial value of Y. Y. X. Is equal to one is equal to zero. Okay so how we can solve the situation The situation resembled with our equation devoured 80 plus B. Y is equal to you. So in this kind of situation how we can solve this by evaluating the integrated factors. Integrated factor is exponential T into a DT. If the derivative factory state. So we get E. The value of pain cases too two Entity DT. So that will be exponential community. Okay now it is time to multiply the C to the power to do two out of the equation. We get It to the 20 day by over DT plus two Y is equal to B Into the -2 T. Into the duty. So we can write this as the over DT into why into E. to the 20 is equal to take so taking T to the right hand side and integrating, we get D O Y E to the treaty is a curtain T. D T plus C sees the constant of integration. So here we get Y O E. To the duty is equal to please feel by a group Plus one plus let us use them boundary condition which is vie, it X is equal to one. Y. It is equal to one is equal to zero, substituting T is equal to one. We get vie sorry, the value variety is equal to one Is you? So that is you And this will be T is equal to 1. 1 x two Plus. This implies sees equal to -1. Let us substitute this value of seeing equation. We get we get by E to the T. Is equal to the square right oh -1 by taking exponential to the to the right hand side. We get why exponential minus two by two into t square minus which is the required solution of the aggression. All these clears your doubt and thank you.


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