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9u(x,t) d2u(x,t) dx2 dt2For 0 < x <1,0 <t <1 With initial boundary conditions:du(x,t) u(x,0) = 0, =5 sin(Tx), att=0 u(0,t) = 0, u(1,t) = 0 Note that the...

Question

9u(x,t) d2u(x,t) dx2 dt2For 0 < x <1,0 <t <1 With initial boundary conditions:du(x,t) u(x,0) = 0, =5 sin(Tx), att=0 u(0,t) = 0, u(1,t) = 0 Note that the true analytical solution is u(x,t) = 2.5/n sin(nx) sin (2nt)

9u(x,t) d2u(x,t) dx2 dt2 For 0 < x <1,0 <t <1 With initial boundary conditions: du(x,t) u(x,0) = 0, =5 sin(Tx), att=0 u(0,t) = 0, u(1,t) = 0 Note that the true analytical solution is u(x,t) = 2.5/n sin(nx) sin (2nt)



Answers

Find the particular solution of the differential equation that satisfies the boundary condition.
Differential Equation $\qquad$ Boundary Condition
$y^{\prime} \cos ^{2} x+y-1=0 \quad y(0)=5$

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

Mhm. Right. So in this problem were given the differential equation to find as x squared -2 sign effects. So the first thing I would do is just think about what is the anti directive. So you have to add one to your exponent, multiply by the reciprocal. And here's the thing that a lot of students get confused on that. The anti drift of negative sign would actually be positive co sign. And the reason why I know that off the top of my head is that the drift of of co sign is negative sign. And that too just goes along for the ride. Um So this is the correct anti derivative. And now what we need to establish is that when X. Is pie, The why value is zero. So what we have to do is is plug in zero for why plug and pie for your exes and figure out what you're see. Value would be Now just a reminder if you look at your unit circle, co sign of pie pies over here and co sign is negative one. So the C. Value when I go to solve this um uh This would be negative too because two times negative on his name too. So I would add two to the left side and then subtract that pi cubed over three. Um And what I need to do is go back to my value for C And rewrite that equation y equals 1 3rd X cubed. Was it plus to co sign of X. Yeah. Uh And then the C value replace it with that two minus pi cubed over three. Um And this is your correct answer because the derivative of this is the differential equation appear and it satisfies the initial condition that one excess pie, The Y Value is zero.

And this problem we're talking about an introduction to the calculation is that you would be doing in a boundary value problem. And that's what we're going to be talking about today. How to go through that process in order to get a final solution. So it's first review what we're given. We have that why Double Prime Plus four y equals the coastline of X. And we're also given some conditions. About what? Why Prime equals. So let's first find the homogeneous part of our solution. Well, we can see that are squared equals four. So if we were to find our by itself would get plus or minus two I if we just take the square root of each side. So what does that tell us? That tells us that Alfa equals zero and beta equals two. So now we can plug these values in to an equation. So we're going to end up with why equals C one times the co sign of two X plus C, two times a sign of two X. So then we want to find these coefficients. We wanna find C one c two. How are we going to do that? But We're going to use a process where we can compare coefficients to find them. So we can say that Y p equals a times of co sin of X plus B Time to sign of X. So now let's plug this into our original equation. Then we would get a cosine x plus b sine x double prime plus four times a cosign X plus B sine X equals the coastline of X so you can factor these or distribute them essentially. And what we find is that B equals zero and A was 1/3. So why in this case would be won over three times the coastline of X? So what we're going to get is why equals c one cosign two x plus C to sign of two X plus 1/3, the coastline of X. So now we have to investigate. Our initial terms were given these initial values or boundaries in this problem, we're told that why prime of zero equals zero, So why prime is going to be equal to negative To see one sign two x plus two c to co sign a two x minus 1/3 sine X Hopefully you can see how I got. Why Prime? Because we knew why. And then from here we can see that C equals zero. We're also told that see, prime of Pi equals zero. So we can say that C two also equal zero. So we get that our final solution to this problem is why equals C one co sign of two X plus 1/3 times the coastline of X. So I hope that this help to understand a little bit more about how we go through the process of understanding these, um, these questions that deal with things such as boundary problems and things like multiple derivatives. So I hope that this made sense, and I hope that you can apply this to your learning in your class.

This question tells us to solve the initial value problem. And then Graff, what we noticed that we have sign of why do you Why his sign of Axe de luxe? We know we're integrating both sides On the left hand side, the integral of sine is negative Coastline y and then same thing. The integral sign is night of co sign and then we have excellent right hand side. If we got rid of the negative signs on the left inside we have coast unwise course on X minus C. We know that we can plug in excess year in the equation. Therefore, what we have is a C that is one. So what this means is that co sign why is co sign X minus one? If this is 1.5, then you can see this is a little bit above


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