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(20 points) A function u(z,t) satisfies the PDEUII0 <I < 2 t20and the homogeneous boundary conditionsu(0,t) =0,2(2,+) = 0,t20Setting u(z,t) F(r)G(t) implies t...

Question

(20 points) A function u(z,t) satisfies the PDEUII0 <I < 2 t20and the homogeneous boundary conditionsu(0,t) =0,2(2,+) = 0,t20Setting u(z,t) F(r)G(t) implies the separated equations F" 2F" AF = 0 and G' + AG = 0_ where is a separation constant _(a) Find all real eigenvalues A and corresponding eigenfunctions F(z) of the eigenvalue problem con sisting of the separated equation for F(r) and the boundary conditions F (0) and F(2) implied by the homogeneous BCs for u(I,t) (Hi

(20 points) A function u(z,t) satisfies the PDE UII 0 <I < 2 t20 and the homogeneous boundary conditions u(0,t) =0, 2(2,+) = 0, t20 Setting u(z,t) F(r)G(t) implies the separated equations F" 2F" AF = 0 and G' + AG = 0_ where is a separation constant _ (a) Find all real eigenvalues A and corresponding eigenfunctions F(z) of the eigenvalue problem con sisting of the separated equation for F(r) and the boundary conditions F (0) and F(2) implied by the homogeneous BCs for u(I,t) (Hint: Iyou need only consider Solve the separated equation for G(t) with the values for obtained in part (a), and then SUm over all of the product solutions F(r)G(t) found to form the general solution satisfying the PDE and its BCs. Find the solution of the PDE and BCs subject to the initial condition u(z,0) = 3ef sin(Tr) 2u:



Answers

For the PDE in Problem 29, assume that the following boundary conditions are imposed:
$$\begin{array} { l } { u ( 0 , y , t ) = u ( a , y , t ) = 0 ; \quad 0 \leq y \leq b , \quad t \geq 0 } \\ { \frac { \partial u } { \partial y } ( x , 0 , t ) = \frac { \partial u } { \partial y } ( x , b , t ) = 0 ; \quad 0 \leq x \leq a , \quad t \geq 0 } \end{array}$$
Show that a nontrivial solution of the form $$u ( x , y , t ) =
X ( x ) Y ( y ) T $$ must satisfy the boundary conditions
$$\begin{array} { l } { X ( 0 ) = X ( a ) = 0 } \\ { Y ^ { \prime } ( 0 ) = Y ^ { \prime } ( b ) = 0 } \end{array} $$

Hello everyone. Today we're going to for carbon number 20 months they've given a startling question. Is violence Yes. One by X in the white equals zero. So be off X equals one bags and cure for lexical zero Integrating factory equals e integral one by X the X so which is equal toe. Here s to Ellen X which is equal to X. Next thing we need to find is by indu integrating factory Go to integral que ethics into integrating factor the X So why equals X Y equals C? Why you cause seeing my ex? This is general solution So why off two equals to do with the momentary condition given for two equals See by two c equals four Why you cause for by X this is the as thick solution and white guys equals minus four by exit square. Thank you

Hello, everyone. Today we're going to start from the number 23 here. The given first order differential equation is X divide equals X plus right plus two. The X so and wondered condition is going like buy off. Barnicle's took 10 so divide by the X equals X plus y plus two okay by X delayed by the X minus one by X in the way it was. Eggs plus two Divided by X. This p affects medicine bags, and this is cure Fixed X plus two by X So integrating factor equals e integral minus one by X the X, which is equal to erase toe minus Ln X, which is equals toe access to minus or which is equal to one by x. So why in tow one by X equals integral one by X into X plus To buy eggs The X so y equals x integral one by X plus to express to minus two, the X y equals X at Lenox minus two plus C x Boundary condition is way off. Barnicle's took 10 so 10 equals one in tow. Ellen one minus two plus e off right, 10 nickels C minus two c equals to 12. So why equals eggs? Atlantic plus two elects minus two. Thank you

Today we're going to solve from the number 17 here The given equation is valued eyes In due course, the square it's plus y minus one equals zero. So why of zero? It was fighting. So why you guys plus one by caused square X in the white equals one by cause the square x here p off Mexico's one because square it's cure fixes equal to one day called square X So integrating factor recalls he integral six square x the X which is equal to us to Panix Why in do it is too planets it was integral into the power Panix in Duke six correx the X right into into the power. Then it's, you know, equals into the power on its plus c. So eat the power Panix into white in tow. The par minus 10 x This one lets see u to the power minus panics. All right, so boundary condition is gonna like y of zero equals five so we can find constancy value for file because one plus see it, the bird minus 10 0 so five equals one plus e in the one see because five minus one, which is equal to four. So Y equals one plus four. Developed by into the power dynamics, which is the particular solution. Thank you.

Hello, everyone. Today we're going to solve problem number 20 here The U. N first order differential location is wide eyes plus two X minus one in the white equals zero. This is P O effects and this is que all fits so integrating factor equals e in the girl two X minus one The it's it is going to eat the products in the X minus one. So why India Integrating factor equals integral cure Fix Indu integrating factor that ISS by indu integrating factor It was individual cure fits into the X So what you'll be getting is like Why in tow bar X into X minus one equals C Well, I equals c different. But you know the products in the X minus one border conditions are like way off one equals toe two equals C diverted by you to the ball one Indo zero So Sequels what he calls to divided by you Do the projects in tow X minus one. Thank you


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