Question

the below is the previous question solution:

1. Recall the following boundary-value problem on the interval [0, 1] from Homework 2: f" =-Xf, f'(1) =-f(1). f(0) = 0, Show that if (Anh) and to this boundary-value problem, λι, λ2 〉 0, λιメÂn then fi and f2 are orthogonal with respect to the standard inner product (.9)J( gr)dr. (You may use the solution posted on the course website, or work directly from the equation and boundary conditions above.) (λ2'J2) are two solutions 2. Find all numbers λ > 0 for which there is a nonzero function f on (0, 1) satisfying f" =-RY, f'(1) =-f(1). f(0) = 0, Also find the corresponding functions f. (Note: it is enough to find an equation which λ must satisfy. It is in general not possible to solve this equation.) The general solution to the given differential equation is (using z as the independent variable f(x) = asin λα + b cos λ. The first boundary condition gives so that we may write f(x) asin Ar. The second boundary condition then gives Since we want f f 0 (note that this means that f and 0 are not the same function, .e., that f is not identically zero; it does not mean that there is no z for which f), we cannot have a 0; thus we may cancel the a from this equation to obtain Thus, if A 0 is any solution to this equation, then f(x) asinAr will satisfy the given boundary value problem for any a. (In principle, a could even be a complex mumber.)

Answer 1

Solutiom 21-22 1 (7)(ス232 (*))dx 0 Γ. 20+ο e-R)「fi@b(n)dn = Ο Thus ,