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Find the general form of the particular solution of the following differential equation: dy dy =I+er dz? drYp = a + bx + cez Yp = ax + bx? + crez Yp =ax + bz? + cer...

Question

Find the general form of the particular solution of the following differential equation: dy dy =I+er dz? drYp = a + bx + cez Yp = ax + bx? + crez Yp =ax + bz? + cer Yp = @ + bx + cre"

Find the general form of the particular solution of the following differential equation: dy dy =I+er dz? dr Yp = a + bx + cez Yp = ax + bx? + crez Yp =ax + bz? + cer Yp = @ + bx + cre"



Answers

Find the general solution of the differential equation. $$\frac{d y}{d x}=\frac{x}{y}$$

Okay. We'll start this problem by rewriting why prime is Dy Dx and this equals to why and then we can move terms around. So let's keep the wives to one side. So I. Y. Dy sorry one over. Why do I? It was one of the access to the other side. So have one over X. Dx. Okay integrate both sides. So on the left hand side we should integrate to natural log the absolute value of Y. On the right hand side we'll integrate to the natural log of the absolute value of X plus C. Now we raise both sides to eat power. Well have Y equals to see X. And that's our answer.

All right, So to find this differential equation or didn't find the general solution to this, we're first going Teoh, turn it into, uh, a linear operator. So that's gonna be D cube minus d squared. Plus D ah, minus one of why is it zero? So the corresponding auxiliary equation p of our is going to be equal to our cute minus R squared plus R minus one is equal to zero. So, uhm, I'm going to attempt to factor this by using by using synthetic division. So I write down the factors. One negative, one one negative one here. And I'm going Teoh trying to factor it using the roots of one. So remember, by rational Ruthie Room. Since this is one here, we only need to look at the factors of this in order to get the roots. So I'm going to try one. So if I do one here, so 11 then that becomes 00 term zeros. 111 Ah, so get zero. All right, so then that means I can factor this Ah polynomial here into our minus one. And then these are the coefficients of the remainder. So it's gonna be r squared plus lips R squared plus one here that's equal to zero. So then now with this, I get r is equal to one And then with this here I get r is equal to r R squared X equal to negative one. So I get are is able to plus or minus I. So that means our general solution is gonna be why of X is equal to. So I start with this route first, So see one e to the X and then I do these two routes next. So this is gonna correspond to a plus or minus eso. Since we have an imaginary it's gonna be co sign and sign S O. C. Two and then the real part is zero. So I'm just gonna go straight to sign of X and then plus C ah, three design of X like so. So this is gonna be our total general solution here

All right, silver problem. 17. You have to find the general solution to this differential equation. And when you have a simple differential equation like when all the coefficients are constants, then you can assume that the solution is in the form of E to the power for constant times the independent variable. We're going to use X for the independent variable, but you can use any variable you want and likewise, we're gonna divide. Derive this. So why prime Miss Movie K Times, Seats of power, Chaos And why don't finally see what's in the case query times if the parking axe and only substitute these into the differential equation. So it's going to be Kase Goro times you to a party next time K time teeth of parking explosive. You need to have heart attacks. Says he wants a zero. We substitute on T to the power tax. So it's gonna be you took architects Times Case carried for us. K post One is he was a zero. We now find their values for K such that this equation equals zero. We know that the left side is never gonna equal to zero since is an exponential function misreadings We're gonna rely on the right side T equals zero and conveniently is just a quadratic equation. So its case growing up last night, case group was came. Plus one supposes Europe and we're going to use the quadratic formula. So negative one poster. Nice a square once, Grace Just one minus four times one size one over two times weren't neither one Folsom lines the spirit of negative 3/2. That's very with omega three. If you was, uh, I have science route three. So in the end, our solutions for K are gonna be negative. One force from Linus. I've rich three over its here, and we just deployed these values into the solution formats. So it's gonna be why, in secret, too, the first constant times e to the power of negative one Linus, I routes 3/2 X and then we add the other constant with it. So it's gonna be plus the second constant times he's a powerful negative one plus for three I over two x And now this curve Yes, I mean this. Generally, we don't consider this a solution, since we have to like, make sure everything is in the real world. We can't have any imaginary values. So before we do, before we go on, we're gonna sports cars. These the turns. So it's going being each of our like that 1/2 x times U to the power of negative. I wrote 32 x and on the right side we're gonna have years and power negative one have X times e to the power of positive I heard 3/2 x and we factor on e to the negative 1/2 excess movie each a negative Excellent tee times See one e t. A power of negative I read three x plus the teeter power constant Iran +32 x And in order to turn thes terms in tow, rial terms working, we would use the McLaurin Siri's for you to the part of acts and sign next and close on xto working out. But it's a lot of writing. So basically the formula and you only through among for us Eton Park I a times a constant will say beta times the independent variable acts If you go to co signed of data X first I times sign of Data X and we're gonna use this formula. It's Ah, make these terms, riel. I guess so. Start with the first term. Using the power. Negative. I threw 32 X It's going to go toe co Sign off, Meg. Never. 3/2 X plus I times sign off. Negative route through with your tax co sign isn't even function, which brings the legs of here is not going to matter whereas the Sinus an odd function which means the negative inside is gonna turn into a negative sign function. And now we do the same for the second term. So it's gonna be e to the positive I threw 32 X Yes, he was co sign of There are three to ax. Plus I Time sign We're 32 X and that we just replaced these terms with these Cosan and sign functions. So we're going to focus on this parts, so it's gonna be see one times co signing off. I'm gonna be substituted for 32 x with C. Since it's gonna be easier to write. So she is going to go toe rule 32 X So it's gonna be co sign of C minus. I times sign with tea for us. C. Two times Kerr. Sign of tea for us. I time sign of team We know foil both of these turns. So it's gonna be See one co sign of tea minus seal on times eyes time sine of t plus See to science coastline tick for a C two times I times sign of C We now combine like terms so the coastline functions and the sine functions So it's gonna be see one for C to of course I'm see Plus negative C one plus e t times I time sign of fake And we're going to simplify this by introducing new Constance. So when you add the two constant numbers together like C one plus C two, well, it's just gonna turn out to be another constant number, which will say C three. Likewise, if you're subtracted to constant numbers together and multiply them by imaginary number, I Well, this whole thing is also gonna turn out to be another constant number, which will say C four and then we substitute this whole thing back here and this will give us our final solution. Actually, So our final solutions can be why secrets at eight and negative x over two times seem stirring, whether number you like. Two years Times Co Sign of through 3/2 x were started shooting t Vanquish North very much. Iraq's and plus C four times Sign of Ruth 3/2 x and, yeah, this is basically adds

Okay, so for this problem let's go ahead and start off by replacing these white terms with our terms instead. So have R squared plus three. R plus two equals zero. From there. We can go ahead and do a little of factoring and we should get our plus two Times are Plus one and this equals zero. And then from this we can extract that are our values are negative one and negative two. With these are values. We can go ahead and build our solution solutions of the form Y equals to see one, eat the negative X Plus C. to eat the -2 x. And so this is our answer to the differential equation above.


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