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Solve the IVP of the inhomogeneous wave equation Ut Uxx =X-t, <x < Co_ t > 0 u(x,0) =x2, ut(x,0) = sin X , ~Co < X < 0...

Question

Solve the IVP of the inhomogeneous wave equation Ut Uxx =X-t, <x < Co_ t > 0 u(x,0) =x2, ut(x,0) = sin X , ~Co < X < 0

Solve the IVP of the inhomogeneous wave equation Ut Uxx =X-t, <x < Co_ t > 0 u(x,0) =x2, ut(x,0) = sin X , ~Co < X < 0



Answers

Find the particular solution of each differential equation having the given boundary condition(s). $$ \frac{d y}{d x}=x^{2}-2 \sin x, \quad \text { when } x=\pi, y=0 $$

Good one. Today we're going to solve problem number 19. The given Pasztor equation is violence plus right planets because seek X plus call. Six. So this is be or fix and this is Kyul fix. So whilst zero equals one, so integrating factor equals a integral panix D F, which is a quarto. See kicks so white in Do Seek X equals Integral CKX. Seek X plus call six DX for right equals Call six in the girl six square it's plus one the X So bye equals call six in do Panix plus X plus c So y equals so next plus Xcor. Six. Let's see because X this is the general solution. So boundary condition is given like y zero equals What? So one equals zero plus zero plus e Very good C equals one. So why equals? Sorry. Next plus X call sex plus call six. This is the specific solution. Thank you

Can its offer home Jesus radiantly at what squared? Plus one is two. So based on this, we know that man there's equal to first minus I should get the wine of ages. You good to see you. One line of X proceed to post under vex. Okay, now it's all for our particular y p is gonna be a deformed you X Ryan. That's close to be ex coast intellects. You know that's you prime sighing likes be prime course because he couldn't don't And we know that you prime and in distributive of sign is co sign of X minus b Prime side of X is he could sequence of X But you don't know your sign of X Your prime, my intellect First he crime coast Finamex was co sign of X you Time co sign of X minus three Prime side of X Mr Holding If he could do sequins of ex co sign of X Hey which this cancers this we get that Are you primacy? Which one? So are you Is he good to X? And then if we have that co sign of X is equal to you Prime sign of X must be fine. Cool Sign of X. By this sign of X, your prime co sign of X might just be time. Sign of X is holding is equal to negative sequins of x kind of x. Then he has to cancel. We left with the prime musical to Tangent of X So we get that Be easy Good to Owen of co sign of X. Okay, so our solution becomes why you got to see one plus x Finamex plus C two votes Ellen of Coastline Rex Times Coastline of X.

Looking at non homogeneous linear equations here in particular, the second order ones. I was a little cheeky. And this problem? Yeah. Uh huh. Primitive of why. Plus why is equal Thio? That's when he used the method Variation of parameters. So the first step is going to be twos. The no. Right. Well, all right. Basically equal to the second derivative. So r squared equal to the second derivative of why On the left hand side way, solve this for our all right here we get the eggs in squared of native. Now, we'll also notice that if we were to did you the negative? Yeah, the square root, the B squared as or a c be less than zero because here be zero. It's like it's not that we just have minus four a c um, both a and see a positive. This means that our solution will have the form. Well, mhm, uh, x times see one to I. It acts in our case. It Yeah. All right. Here. Uh huh. We want to use the additional parameters to look at a particular solution, which means we have constant. You really need to Yeah, Yeah. Mhm. Yeah, Yeah. So you find the second derivative of this particular solution. Absolutely warning. Oh, yes. Uh huh. It's Yeah. This he had together legal zero. So mhm minus the next for the first. A particular solution. So we take the really right, He, um minus. Yeah, I am minus you one prime, um, sign legs minus you one so set of X. So if we substitute that into our equation, um, that what can't square, which is also went over. I was saying Right, so alright, it that way. Um, since our solutions have co science and easier for simplifying. So we get, um, please then and do behind We end up with YouTube times the Senate X minus you. One prime times sign of X is equal. You went over because since square necks. So now we have a system of equations. We consult. So you have you to prime a sign of X minus one prime kind of X is equal to one of workers sine squared. Also, have you one prime the Senate X plus YouTube prime sign of X equals zero. So we're going to solve this to find you wanted you to from the second equation we can just get at you want Time is equal to negative You to prime times Fine of X over coastline of X. We substitute that into the first equation. Get YouTube. I'm a sign of X plus you to prime. I am a squared X over because son of X is equal to one over cause I in squared x eso we can multiply the first expression by cosine over cosine so that it's all, um everything's over coast an index, then, which means that we can multiply both sides because X that gets through the square on the right hand side, we can also bundle up was on the right hand side here to be There are a few to, um, multiplied by close hand squared, plus science where right, which is just gonna be equal to one. So what we end up with is third of YouTube is equal to one over percent of X we can put it into we had for you wanna prime, uh, to get that? That is going to be equal to negative. So that was Synnex. A very close friend of ex um has won workers have X will have negative sign of X over the science quarterbacks. Let me want to integrate these. Okay, so we integrate YouTube, YouTube crime. So just maybe even the integral of one over, because it lacks, which is equal. Thio, you can't x on. This is equal to Ellen. Uh, so you can't have X plus tangent of X? I think it was in chapter seven that you see, this s if you're wondering how that worked out. Go there. I do the same thing. Find an excellent yeah, X if you do a substitution Well, that co sin of X b. Okay, so just a co sign of X is equal to W and, of course, sign of X. Native son X becomes VW in this integral is e missed the negative out front there. And so this interval ends up being just went over that b squared he w which we know how to solve. Yes, that's just gonna be equal to negative one over w. This means that you won is equal to negative one over there. That's it, X. So if we substitute this in to our particular solution, get negative one was the natural law of he can't do. X plus engine 10 x I am sign of X and therefore the general Solution Planet X minus one plus natural log. You can't x plus engine of x times sign of X.

This question asked us to solve the differential equation. We know that we have teeth. Data over DT is too secret Data divide by Breda e to the T Square. No, What we know we need to do is we know we want all that they doesn't left inside and all the cheese on the right hand side. So multiply both sides by data to get rid of that data on the right and bring it over to the left. Simple five. We know one over seeking is the same thing as co sign and then this. We can write this with the negative exponents like this. No, we know we're gonna be taking the integral of both sides. As you can see over here, this is the first integral sign this second integral sign. You can do this by using the formula for integration by parts. All right, it over here. Given the fact that you is negative, cheese squared. I get to d t. As now you're to t d g is do you and then t d t is negative 1/2. Do you? We know we can use integration by parts to do negative 1/2 times the integral of heat of the U. D you, which gives us negative 1/2 feet of the you pussy. Now, last step, you needs to be substituted in so back. Substitute. Negative 1/2 e. Remember, you was actually negative, Chief scored. So we're literally just plug in your end and this is our final solution.


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