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Y = f(x,y) = g(x)h(y) is a Ist order ODE that isdx 9x = 2cos 3t is called aThe equation 3order; (linear/nonlinear) (homogeneous nonhomogeneous): ordinary differenti...

Question

Y = f(x,y) = g(x)h(y) is a Ist order ODE that isdx 9x = 2cos 3t is called aThe equation 3order; (linear/nonlinear) (homogeneous nonhomogeneous): ordinary differential equation. A solution to an ODE is y = 3-Ce 2r_ Here, Ce-2x is called the term because it approaches zero as x approaches infinity.

y = f(x,y) = g(x)h(y) is a Ist order ODE that is dx 9x = 2cos 3t is called a The equation 3 order; (linear/nonlinear) (homogeneous nonhomogeneous): ordinary differential equation. A solution to an ODE is y = 3-Ce 2r_ Here, Ce-2x is called the term because it approaches zero as x approaches infinity.



Answers

Find the general solution of the given second-order differential equation. $$3 y^{\prime \prime}+y=0$$

Okay, well, given x squared y double prime plus X y prime plus X squared minus one times. Why is he good to go? Were asked to find a general or asked to find the solution using our power. Siri's So we know that X or y double prime a ticket to end summation from energies to to infinity. And there's an A minus one day and extra curve and minus two. And then we have X times, the summation of energy equal to one infinity of end times a n extra perv. And when it's one plus X squared minus one time to some from energy to go to infinity of a in sex and this is all equal to jump. Okay, so we're gonna have to be in Texas. That's really fix this all in terms of energy, it's too. So we get the summation, Mom, And if you could see to the infinity of and there's an minus one and extra gun us see, we have a one of x plus summations from anything too Infinity and a n X and class summation from Infinity. And she was too in minus two extra and minus a not manage a one X So these to cancel who's Ah, I could start factoring out our extra. And yet this is in terms and minus one in us and A n A and modest too minus Ann. Well, your ex and might I say not as well. No, not there. Who we missing are a Not so we have this holding minus a knot is equal to Doc. Okay, we can simplify this further. Get the summation from energy. It's too to infinity of and squared minus one and plus a n minus, two times X to carve and minus a knot. Is he good to go to get on A Not because you got to go in that a m is equal to a and minutes to over one minute and scored. Okay, Okay. So if we tried to solve four a not Honoria up to Mmm mmm. Mmm. They're only good till it gets offer Adri to have a 31 over one of his three scored 81 a four or a five is equal to since a four is gonna equal to 00 actually, all over and when even they're gonna equal to zero. So a five minute one. Over what? Minus five Squared a three. You ready? Not we get one over one. Witness. Five word when one is three squared, ate one. It seems like we're gonna add on our previous values. So our general solution is why is equal to see X C summation from anything that's too to infinity. Ah, X two and minus one. All over. One minus two and minus one to borrow a suit. I don't know about that. The ball times one minus three squared.

Okay. So for this problem let's go ahead and use our general substitution that we used to eradicate these white terms. So we'll have our R cubed plus three R squared plus three R plus one equals to zero. And let's go ahead. In fact this and we do we should get our plus one Cubed and equals to zero. And from this we can derive that are our values are negative one but it's repeating right? So it's repeating twice. So in our solution which is going to be Y equals we'll have K one E to the negative X Plus K two X. Because it's repeating. Eat the negative X. K plus K three X squared, pizza negative X. And we're adding this X squared and X component simply because it's repeating. And so in this case this is our answer.

Okay, so start off this problem. Let's go ahead and replace this. Y prime. Why white old prime and white terms with R squared minus three are Plus two equals 0. With that we can do a little bit of factoring, still have ar minus two times. Ar minus one equals zero from this. We can derive that the roots in this case must be one and two. And so with that we can build our solution. So solution is going to be Y equals two K, one E to the X plus k, two, E two, the two X. And so that's our final answer.

These questions for the topic relating to separable person on the differential equations. All right. First you need to separate the equation. You keep one on the left hand side and on the right hand side. So that means you have to Y E Y. E. Co two negative sign of three X. Divided by ghosts. I keep of three X. G. X. Right. And then you take the intolerable size. You're going to have tried ey echo to the intro of negative sigh of three x. divided by goals. I keep three X. D. X. Right. And from the left hand side you can easily get I swear plus I'm costing. But I want to let my content on the right. I don't write down On the left hand side on the right hand side and I have the of gold side of three X. of course I keep up three x. and then because the derivative of co sign is negative sign. Right? So you have to use the chain rule also. So it's gonna be one third because you take the three outside. All right. So this is gonna be like that and it's equal to what? So so it's equal to whites when equal to 1/3 course I to the negative three plus one. Right. Three X. Divided by negative three plus one. Because this is it Yeah it's very easy to compute and then you get White Square Echo two negative. So this is negative two. So it's gonna be negative 16 of gold size square of three X. Blossom constant. Right. And that's it. So your solution is Why Square Plus 1 6? Of course, I square of three X equal to some constant. So this is an implicit form lisa solution, I said.


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