1 Point Find The Eigenfunctions And Eigenvalues Of The Differential Equation Day Ag 0 D C2 Y 0 Y 0 0 Y 9 1 (67.52 KiB) Viewed 30 times
(1 point) Find the eigenfunctions and eigenvalues of the differential equation day + Ag = 0 d.c2 y(0) + y (0) = 0, y(9) = 0 For the general solution of the differential equation in the following cases use A and B for your constants and list the function in alphabetical order, for example y= A cos(x) + B sin(x). For the variable A type the word lambda, otherwise treat it as you would any other variable. Case 1: X=0 The general solution is Ae^(sqrt(8))+Be^-(sqrt(8)) The initial conditions lead to two equations in two unknowns in a and b: 0 = 0 0 = 0 Solving this system for A and B we get A= 0 B = 0 Case 2: < 0 The general solution is Asqrt(lambda)e(sqrt(lambdax))-Be(sqrt(lambdax)) The initial conditions lead to two equations in two unknowns in A and B: А = 0 B = 0 Solve the first equation for A: A = 0 Plug this into the first equation, taking out the common factor of B, and you get B 0 0
. 0 If B+ 0 then tanh(sqrt(lambda)) Choose the graph of Vand tanh (9/1) ΠΑ. Α ОВ. В C.C D D 313 313 313 313 FE -1 2.0 2.0 2.0 A B С D From the graph there are 0 nonzero eigenvalues for X: 0 (Your answer should be accurate to 9 places) Sturm-Liouville guarantees infinitely many eigenvalues, so we go on: Case 3: >0 The general solution is Acos(alphax)+Bsin(alphax) The initial conditions lead two equations in two unknowns in a and b: A+8B - 0
Sturm-Liouville guarantees infinitely many eigenvalues, so we go on: Case 3: >0 The general solution is Acos(alphax)+Bsin(alphax) The initial conditions lead to two equations in two unknowns in a and b: A+8B = 0 A+2B = 0 Solve the first equation for A: A = 0 Plug this into the first equation, taking out the common factor of B, and you get 0 0 Therefore tan(lambda) = The first four nonzero eigenvalues from this equation are: 0 Solution List the first three nonzero eigenfunctions in order of increasing eigenvalues 0 Here are graphs of the first four nonzero eigenfunctions on (0,9). 313 F1
Solution List the first three nonzero eigenfunctions in order of increasing eigenvalues 0 Here are graphs of the first four nonzero eigenfunctions on (0, 9]. 313 -5 List the functions in increasing order by their color, green, red, blue: 0
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