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consider a boundary value problem y"+ky=0, y(0)=0, Y(pi/2)=0. is it possible to determine the values k for trivial solutions. (b) non-trivial.

Differential Equations
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\[y''+ky=0 \implies y=A \cos(\sqrt k t)+B \sin(\sqrt k t)\] From here you can plug in the initial conditions
Since the boundary conditions are homogeneous, your non-trivial solution (the case \(k>0\)) will be a Fourier sine series. For \(k\leqslant0\), you'll get trivial solutions.

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