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The given differential equation is: [ x y'' + y' = 0, \quad 0 < x < \ell ] with boundary conditions: (y(0)) bounded and (y(\ell) = 0). --- Step 1: Solve the homogeneous equation The equation can be rewritten as: [ y'' + \frac{1}{x} y' = 0. ] Let (y' = u). Then (y'' = u'), and the equation becomes: [ u' + \frac{1}{x} u = 0. ] This is a first-order linear equation in (u). Its integrating factor is (\mu(x) = e^{\int (1/x) dx} = x). Multiply both sides by (x): [ x u' + u = 0 \implies \frac{d}{dx} (x u) _______ _____ ______ ____ __________ _____ ________ ___ _____ _____ __________ _____.
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