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Question

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Answers

Determine an integrating factor for the given differential equation and hence general solution

In this question we have to find a linear homogeneous differential equations with constant coffee. Since whose rules haven't given us M equals 000 minus two minus 23 out and minus three. Health will first consider the first three zeros that have given us routes since Amy called zero. You can see the M plus zero zero again and plus zero zero again and plus zero. So you can see that Michelangelo and will multiply all these three. We'll get em kiwi called zero. Now let's just see the second one that is equal to -2. And again in cold minus. So if M equals minus that, we can see that M plus triple zero. And again, because the rules are minus and minus. So we multiply these two. That is plus 1 10 plus two. We'll get down to +00 plus two will be Now for the hard part that is m equal to three out and again, simple to minus three out. So for chemical three out and Ministry out of Visual and for me called minus helpless three out of So we'll multiple at this represents that is -3 out into the first three out and we'll get 0-00. So we'll get when you multiply these two will get them square -9. I spell it and we know that I ought to swear you called-. So we'll get em effortless 90 collagen. Now again will multiple all these 3% that is this one this one and this one and we'll get zero that is thank you And two members to elsewhere. And um start last night. Now we'll just open the brackets plus two. Open this square square one. That is m plus two holes. Whether it will get M squared plus four plus four. Now again, we'll open this bracket, keeping them to we'll get em fourth plus nine M squared plus four M. Two. Statistics name plus 26.0. Now again. Well, open up the record for the final time. That this will get empty about seven plus four and six. They're starting them to fight the strategic 4 36 M two Q. Is equal to zero. No, just try to remember this. We know that when if this is the given different allocation, we try to replace by double death by M square and wider by him. And why simply were one to get out of the equation and then solve for the roots and get the differential equations. Also the different different person. Similarly, we'll go and solve this one, but we'll go reverse this time. But if it's uh auxiliary question is this one, what will we tell modernist different person. This will put And to those seven years. What? You were seven and into a success right? Or 6 25 25 24 hours prior to the fourth and M two or three years? And this will be there. Our homelessness differential equations with Western conference whose rules will be this that is mhm 000 -2 -2, 3 out of zero, and this will be the auxiliary person. Yeah. Mhm.

Here a second daughter differential equation is given. And we have to find a general solution of this differential equation, as we can observe here that right inside of this differential equation is zero. Therefore this equation is a homogeneous equation. So for how to solve homogeneous equation, we first find an auxiliary equation. And auxiliary equation is found by replacing the term day square by by D. T square with D square and divide by DT with deep. So auxiliary equation will be auxiliary equation of this differential equation will be capital D square minus nine. Candy is equals to zero. No, we have to find the roots of this equation, so D into D minus nine physicals to zero. Therefore, route of this equation, this auxiliary equation will be zero and nine. So here the term Jiro and nine. We have to put it in form of R one and R two. So as in the problem, it is mentioned that are one is less than R two. Therefore our win will be zero and R two is equals to nine. So these are the rules of this Angeleri equation. Now, with the help of the rules, we can write the complementary function for this differential equation. So complementary function Cf is seven. He raised to the power our win T since very early here in this problem is T. Therefore we have to use T. Here plus C. Two equals to the power R. Two. T. So we can put the value of R. One and R. Two. So this will be C. One ear is to the power zero, multiplied by T. Plus C. Two equals to the power 19. So this uh solved if we hear this time he raised to the power zero. T. Will be erased to the power zero and he raised to the power zero is one. Therefore this term will be one into 77 plus C. To be rich to the power 90. So this is the complementary function for this our differential equation, since this is a homogeneous differential equation. So for homogeneous differential equation, general solution that Gs is cf of the function only. So we need not to calculate particular integral for this solution for this particular question. So general solution will be why is equals two seven plus C. Two he raised to the power 19. So this is the answer for this problem.

In mathematics a differential equation is an equation that relates for normal functions and their derivatives an initial problem but a problem or I. D. P. Is a differential equation along the appropriate number of initial conditions. So you have first everybody get vacations you over the X equals X. Sine X over Y. Can you stay operable techniques or we have, why do I call X M X dx shit. Then we integrate outside evacuation from the lap to the right side of vacation. X and X dx using the integral part. So it calls XB from SAn X. So we have negative X. Course and X minus the integral negative cause I'm X dx we have no that integral of negative causing X. Dx is equal to negative sine X. So we have equals two negative cause an X minus minus sine X. Simplify it. We have negative X. Cause an experience. I'm X add Afghanistan today solution. I got an ex course and experience an X plus C. So to that right up a condition we have Y squared over two. The integral. Why do I see sequence twice credible shit at this plus time plus sine X policy. So now we have Y squared over two equals negative X plus and X plus and X plus average quad three construct C or C. Over to So again across multiple better. So we're boycotters negative to expose and expressed his and express E. Re bend the initial value we have Y. Of zero is equal to negative one substituted to the solutions of differential equations. We have negative one calls negative two times zero causing zero plus two signs zero plus Z. So we have seats equals zero. We shall become zero. And also this 10 substitute today end the occupation here we have now. Why calls negative two X coffin expressed to us and X men's one

In this question were given a differential equation by by squared plus two, ex wife, the x minus x squared The Y is equal to zero. So I will expand this equation by square D X plus two X dx times y minus x squared Dy is equal serum and I can write this S D x squared which yields us Y squared dx plus why T X squared minus x squared dy is equal syrup. And I can write this. I can divide this by y squared which gives us why the x squared minus x squared tv over Y squared is equal to zero. And this is the integrating factor. Me why is equal to one over y squared. And if you closely observed this expression this is the derivative of a quotation which is derivative of X squared over why? Which gives us why times the x squared minus X squared dy over Y squared. So based on this we can right D x plus D off X squared over boy is equal to zero, which means bye integrating. We can right X plus X squared over White is equal to see. So this is the solution of this differential equation


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