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Find one solution to each of the following differential equations by trial and error, and then find second linearly independent solution using reduction of order. F...

Question

Find one solution to each of the following differential equations by trial and error, and then find second linearly independent solution using reduction of order. Finding both solutions by trial and error is unacceptable By computing their Wronskian, show that your two solutions form fundamental set on (0) fv" r(r + ')v + ( +2)0 =0 [ =(,0) (b) Ty" (r + I)v' + "2v =0 [ =(0)

Find one solution to each of the following differential equations by trial and error, and then find second linearly independent solution using reduction of order. Finding both solutions by trial and error is unacceptable By computing their Wronskian, show that your two solutions form fundamental set on (0) fv" r(r + ')v + ( +2)0 =0 [ =(,0) (b) Ty" (r + I)v' + "2v =0 [ =(0)



Answers

Verify that the given functions $y_{1}$ and $y_{2}$ are linearly independent solutions of the following differential equation and find the solution that satisfies the given initial conditions.
$t y^{\prime \prime}-(t+2) y^{\prime}+2 y=0;$

$y_{1}(t)=e^{t}, \quad y_{2}(t)=t^{2}+2 t; +2$
$y(1)=0, \quad y^{\prime}(1)=1$

So we have this differential equation here and we want to find solutions of the form. Why have access with you actually are. First we need to find wife time. This is going to be are actually ar minus one. Now we have why Double prime of X is equal to our times. Ar minus one x to the R minus two to here. So now we substitute these into our equation here, so get two x squared times are times ar minus one Times are to the extra the R minus two plus five x r x to the R minus one plus X two br is equal to zero So we simplify this So we have X squared times actually ar minus two. This will multiply, becomes just x two They are so I go ahead and multiply this out as well to R squared minus two are times X to the are plus again Similarly here this times this is just gonna be exit er we have plus five are axity R plus actually are is equal to zero. Now we're going to factor out and exit the are from all three. All these terms of actually are. And then what's left? We have to r squared minus two are plus five are +100 Um, we know this cannot be even zero for all X greater than zero. Um, so then that means we have to solve, uh, this here to be equal to zero. So that's two r squared and in minus to our past five. Are that's gonna be plus three are plus one is equal to zero. Um, since everything all the signs are positive here, we're just gonna have to our and then our, um one and one. Is this going to be to r plus one and are plus one like that? Right. Um, so we get our and then to artistry are and then to r squared. And then one, this is equal to zero. So this gives us to, um, solutions. This gives us negative 1/2 and negative one. So our solutions of that form are going to be extra negative. 1/2 and x, the negative one

Okay, so for this differential equation, we need to find solutions of the form. Why is equal to Exeter? Are so then why Prime is going to be our X to the R minus one. And then why double time is going to be equal to our times? Ar minus one times, actually. Ar minus two. So now substituting these into this differential equation Here we have two x squared. Our times are minus one times times X to the R minus two plus five x Times are times actually ar minus one. It's plus exit E r is equal to zero. Um, we can simplify this, uh, so this x squared and this actually ar minus who becomes extra the are and then the two and the arse. Times ar minus one. I'm going to turn that into two R squared minus to our Okay. Then move that. You know that a little here and then similarly, here this x and X R minus one becomes our or extra that are so we just get five are times actually are plus X to the are is zero. So now we can see. Um, we can, um, back throughout an extra are from all three terms. So we have. Actually, there are. And then what's left over is to our square minus two are plus five r plus one is equal to zero. Since XLR is not gonna be equal to zero for X greater than zero. Now we just need to set this part of the creation equals zero. So I get to r squared and then minus two r plus five are that becomes plus three are plus one is equal to zero. Okay, so, um, here, since we only have one here, but we have to hear, um, we can factor into to our and then that's just gonna be one here and one here, since everything's positive will just have positive there. Okay, so this does work out. So too are, um is R squared then to our and then our is three r, and then one is one. Okay, so our two solutions are going to be negative 1/2 and negative one. So we're going to have X to the negative 1/2 and then X to the negative one. Eyes are to Linnean independent solution. So our general solution Why? See, that's gonna be c one c one x to the negative 1/2 um, negative 1/2 plus c two x the negative one.

So we have this differential equation and we need to find solutions of the form Why X is equal to eat to the Rx. The first will find why prime is equal to r E to the r X on why Devil Crime of X is equal to r squared each of our X. So fucking, uh, these into here we get that, um, we have our square into the Rx minus 36 e to the R X is equal to zero, and then we can factor out he to the Haaretz. From all in terms we have eaten our X and then r squared R squared minus 36 equals zero. Since either the Rx can never be able to zero, then that means we have to set R squared minus 36 equal to zero. This is a difference of square. So we can back to this like r plus six and R minus six like so, does he go to zero? So there are two solutions are just going to be plus or minus six. So you do the negative six and either the positive six, so are two solutions or two, or solutions are gonna take the form you know, the six x or e to the negative six X, like so

So we need to find solutions to this different situation of the form. Lie of X is equal to either the Rx. So we need to find why Prime, which is going to be able to Ari to the Rx and why Double prime of X is equal to r squared E to the R. X. Next, we substitute these into a differential equation here. So we get r squared E to the R X minus 36. Uh, sorry. 36 e to the R X is equal to zero. Then we can factor out and eat of the Rx from both of these trips. Either. The Rx R squared, minus 36 is equal to zero. This eat of the Rx cannot be equal to zero. So therefore we only need to set this part of the equation equals zero that's R squared minus 36 is equal to zero is equal to zero. So r squared equals 36 so that our is going to be equal to plus or minus six. So six and negative six. Therefore, our two solutions are gonna be equal e the six x either negative six x S o r. General solution. Why Sea of Acts is going to be equal to is able to see one e to the six x plus c two e to the negative six x and that's these. There are solutions.


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