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Consider the ODE:y" + 4y cos(wt)Where w is a positive real number. If w is a positive rational number (ie W is & number of the form for positive integers a...

Question

Consider the ODE:y" + 4y cos(wt)Where w is a positive real number. If w is a positive rational number (ie W is & number of the form for positive integers and n), and not equal to 2, show that any solution of the above ODE is periodic Hint: You 'Il find that the solution is the sum of two periodic functions with different periods; try multiple of those periods If w 2, find the general solution and show that it is unbounded as t 3 O0. In other words, if $(t) is any solution, show tha

Consider the ODE: y" + 4y cos(wt) Where w is a positive real number. If w is a positive rational number (ie W is & number of the form for positive integers and n), and not equal to 2, show that any solution of the above ODE is periodic Hint: You 'Il find that the solution is the sum of two periodic functions with different periods; try multiple of those periods If w 2, find the general solution and show that it is unbounded as t 3 O0. In other words, if $(t) is any solution, show that given any M > 0 and N > 0, YoU can find a t 2 N with |d(t)l 2 M_



Answers

Find the solution to the Cauchy-Euler equation on the interval $(0, \infty) .$ In each case, $m$ and $k$ are positive constants. $$x^{2} y^{\prime \prime}+x y^{\prime}-m^{2} y=0$$

So here we need to first solve the initialization. That's gonna be our square. And then we have a plus. A one minus one are term. So anyone here is just one. So we have one minus one are zero r. Then we're gonna haven't plus a to A to here is negative m squared. So we're gonna have minus M squared. Is he? Will this euro this condemned factor into r equals R minus M and R plus M is it with a zero? So we got that are is gonna be plus or minus m. So our general solution it's gonna be y of x equal. C one exit e m plus C two x to the negative m

Now in this problem, we are given of the differential equation, which is y double fry empty plus K square y all 50 is equal to zero on that three parts Human integration. In the a part, we have to verify that folk A is equals to one. The solution of this initial very problem will be why you'll be is records to see one scientist plus c. Tokyo Sainty. For that, I'll be finding why Prime Plea from this equation and then by prime avoidable friendly. And then I'll plug in the value here so calculate will calculate Viper empty it. It will be considered differentiation of dysfunction. So see one, cause I ain t on here. Rigel minus Cito signed. No y double fry Empty will be close to minus C to Sainty mine. Yeah, it's see one here. It's not CDO. See one sandy and see Tokyo Sainty. Now I'll be blocking the values in this differential question. For that, I'll be considering the left side off the differential equation. So why double frame, please is one so minus Siyuan Scient E minus C Ducasse nt on bless. Here, here is one. So one times one square is also one. So we're left with my off the only So we're going to block the value of y often here. So place see one Sainty plus c Ducasse Sandy this term and establish cancel out on this time in the struggle against low. So you get zero and that is a call to the right side off the division equation. That means we have verified that this is the solution for the given initial value problem. In to be part, we have to verify if K is equal to two in the solution of the differential equation will be see even signed duty less see took a saint duty again. We'll follow the same procedure. First of all, we calibrate by primly villa vehicles to see one consigned to three times to minus C two times same Toby. Times two on y double prime tea will be calls to two times Siyuan signed to be with negative because the differentiation off and two times two is four. So it will become four because a different vision off societies minus scientist and consigned to level B equals minus two times Scientology. So too, was already there two times minus two is minus four. Now here we get four times, see to consign duty. So now again, with the plug in the values off by prime day in the differential equation for that again I'll be considered and left side of the question. So why double frame, please? Is one. It's simply about the valleys, so four minus four C even signed duty minus four games. Cito Hossein Duty. Bless K Square. Biosafety case queries. Here is case to showcase. Gotta be four four times of io for two. So this is my office. Soc even signed duty plus c took assigned duty. So after simplification will get minus force even. Consign duty minus four C two Sorry. Here it is. Minus sign toe here minus four c took assigned duty plus four C one ST duty less C two, Four times. He took a saint duty this time and this time will cancel out again this indestructible casserole. So we get zero on this is a cold right side off the question of the differential equation. So finally we can conclude that four k called to the solution of the differential equation will be this one. This equation. No, in a C part of the gration. So see part we have to verify not very favorable to give the general solution for the differential collusion which was wide double prime tea plus k square. Why off equals +20 We have to get a general solution of this differential equation and then we have to verify also So the general solution will be equals two. Why off b is it calls to see one sign Katie blessed see took a saint duty. So because in Q t where K is a constant greater than one again will be calculating Why Prime d So why Prime table vehicles? To see one Hussein Katie Dymski minus k times, See to sign Katie Onda y doble prime tea will be called to minus K square. See, even sign Kitty on here you get minus K square. See two cor ST duty. So there's a relative opened. I'm going to plug these value in the differential equation. So for that, I'll be concerning the left side off the differential equation. So why prime days this one, simply or to pluck the value minus k square? See one sign Katie minus Kato. See? Sorry, k squared CDO hossain duty and bless gay square times y 50. So I have to use this one. See one sign. Katie, Let's see to co sign Katie just before we were due to simplified Soak a square, See one sign Katie minus k square. See? Took Assane duty. Bless Gay square. See one signed Getty Bless Case square. See to co sign Kitty to this term it on this terminal cancer alone on this one. And this time we'll cancel. So you get zero. And that is a call to the right side off the equation of a differential equation. So hence were verified that this is a general solution for the given initial l a problem. So I hope you like the problem. Thank you.

In this problem. We want to find a solution to the differential equation. Why double prime equal to X to the end. So to do that, we're going to take the indefinite integral of each side. So we have a wide prime is going equal the integral of X to the N D X, which is going to be in plus one x to the in plus one plus c at some concerts. See one. So now that we have that, we can dio why we can integrate this side again and get why is equal to the integral of in plus one times X to the n plus one plus C one All of this integrating with respect acts So this going to be equal to we're gonna do the products role or the, um, the power role again. So we get in plus one times in plus two times X to the n plus two plus C one times X plus some constant seed to. And then we get that This This is our final solution for our differential equation and the solution is going to be valid for any X In the real numbers sense we can plug in any X to these powers. In addition function, there's no stipulations on what X needs to be so X is in The real numbers, or X is between negative infinity to infinity.

So you solve this problem here? Where first going to solve the additional question. That's gonna be r squared. And then we have a plus a one minus one. Our time. A one here is the negative two M minus one like that. So we're gonna have minus to em. Um, some minus two M plus one and then minus one. So I'm just gonna be minus two m r and then we have Oh, sorry. This should be a plus. I'm squared. So then plus a two, which is M squared, is equal to zero. This can factor into n minus R. Well, sorry. Ar minus M squared is equal to zero. So we get that are is able to em with a multiplicity of to. Okay, so then our general solution why of acts is gonna be go to X to the foreign started. See, one adds to the M plus C two and then exit e m times Ellen of X for that second multiplicity


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