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Merve S.

Differential Equations

6 days, 20 hours ago

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Could you help me to solve it?

Hello. The question is taken from differential equation and we have to solve the differential equation. And also the boundary conditions are also given to us. So in order to solve it we need to first solve this general solution of the left hand side. So nine Y. Is equal to zero. And we need to evaluate a complementary function. Okay so in order to evaluate the complementary function, Y prime is equal to Dy over DT Let D over DT is equal to capital in a similar way the square over DT square is equal to capital D square. So from here the square minus 60 plus nine is equal to zero. This implies D is equal to three. Common three. So complimentary solution will be Y is equal to a into the three T plus Bt E to the treaty. This is just a complementary function. Okay now we need to evaluate the particular integral particular in T. V. is equal to 18 T. Into the power treaty plus please. Two E treaty minus 90 plus three. Do I did buy d minus three whole square. Okay so let us always let us uh take the minus three whole square in India is really to eastern in the denominator of Eastern Divided by D -3 whole square plus two T. To the power zero into the treaty divided by the minus three whole square minus 90 Divided by D -3 whole square plus. So u minus three over into the zero deal Divided over the -3 whole school. Okay so let us all one by one. So for first time that is 18 T into the treaty divided over the minus three whole square resolution of the storm is equal to first into the treaty keeping the D. S. D. Plus three. So 18 D. Into the treaty divided over D. Square d. Plus 3 -3. So that will be like 18 into the treaty integration of T with respect to the square. So that will be teach you go well first two and then six. So this will be three Which is equal to three TQ Into the Treaty and 2nd Thomas To T to the Power zero. We do the treaty the I did over The -3 Whole Square. So that will be equal to mm two eight of the treaty divided over the square. Anti to the powers you. So that will be to into the treaty. T squared divided by two will be cancel out. So that becomes the square into the treaty and for third down take minus three homers so minus three old square. So that becomes 90 divided over nine Into 1 -3. Over three. Hold square. Using the binomial. Um Yes using the binomial terms we get one To do world three into T which is equal to T -2/3. Okay and next the Tonys three. He to the reality Over D -3 whole square. That is fine. So this becomes 3/9 which is equal to one over T. So creo if we substitute it is 9 19. Oh they define that is one of all three. So substituting this value in the creation and we get value of particular interest. Particular integral is equal to treaty. QB to the treaty plus the square into the treaty minus T plus two by three Plus one x 3. So that will give us value three. Did you? It or the treaty plus days where eight of the treaty minus T plus one which is the equal value of the particular integer. Now the next solution for the situation is why A. U. To the treaty close bt you do the treaty plus three teach you you do the treaty plus T square into the three T minus D. Plus. Now using the boundary condition in order to evaluate the value of the N. B. Okay so first boundary condition is Y. AT is equal to zero is equal to two. So I'm using those boundary conditions why At P is equal to zero is equal to two. So why At zero will be equal to only a. And this is one Is equal to two. This implies a is equal to one. And the second power tree condition can be evaluated by by brian so that will be three. A. Into the treaty plus be exponential treaty Plus B. three b. P exponential treaty yes nine t square exponential treaty plus nine T. Cube exponential 30 plus to be exponential 30 Plus T Squared Exponential Treaty -1. So y prime at zero Is equal to given it. I think that is 11. So that will be equal to three A plus b minus one. All other times housing this implies and A. Is equal to one. So B will be equal to three minus 1 to 2. B is equal to nine. Okay so the final solution for the situation will be E. E. To the treaty plus B. T. B. So substituting the value of A. N B. A. Is one. Inbee is 90. You do the treaty yes. Treaty square D cube B to the treaty. Yeah plus the square into the treaty minus T plus one which is the required solution for the given equation. So hope this clears your doubt and thank you.

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