Question

(1 point) In this problem we find the eigenfunctions and eigenvalues of the differential equation B+ iy=0 with boundary conditions (0) + (0) = 0 W2) = 0 For the general solution of the differential equation in the following cases use A and B for your constants, for example y = A cos(x) + B sin(x)For the variable i type the word lambda, otherwise treat it as you would any other variable. Case 1: 1 = 0 (1a.) Ignoring the boundary conditions for a moment, the general solution of the differential equation is y(x) = Apply the boundary conditions to the general solution to obtain two equations relating A to B: (16.) Solving this system for A and B we get Case 2: i<0 (2 a.) Ignoring the boundary conditions for a moment, the general solution is y(x) = Apply the boundary conditions to obtain two equations relating A to B: (2b.) Assuming A and B are not both zero then, solving these equations, one can eliminate both A and B and derive the following formula for vi in terms of hyperbolic functions (cosh, sinh, tanh, cochetc.). v=i= (2c.) Choose the graph of -1 and tanh(2 ) ОА OB From the graphs there are negative eigenvalues. They are (Your answers should be rounded to 3 decimal places.) Sturm-Liouville guarantees infinitely many eigenvalues, so we go on.

Answer 1

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case 2: ano Then, d2y +2y=0 ) which gives the solution for the forem grees y (W = A cos sã x + B din toa No, y (x) = - Ata sindana 3.57 costa a Yoo) + yiroq=0 . 5 TA + BJT = 0 - 0 Atsa, (2) = 0 = Aros 276 +B sin 2.1770) from the co-efficient defemminent of @und na 1 = (sinzta – Jacosztá) Casara sinara ਰਿਗ mm zes ,& meਅ ਰੀਪye su kry... we put sinato in corsora = neon tan ain= from the graph wegle that the first four positive einen values are. $2.135, 13.755, 15.406 Veggs = 1.461, :1.948, 23259 2044 Scanned with CamScanner: