Word Count : 457

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We are asked to solve the linear homogeneous differential equation: [ \sin x \frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0, \quad x \in (0, \pi). ] First, rewrite it in standard form by dividing through by (\sin x) (valid for (x \in (0, \pi)), since (\sin x \neq 0)): [ \frac{d^2y}{dx^2} + \frac{1}{\sin x} \frac{dy}{dx} + \frac{1}{\sin x} y = 0. ] This is a second-order linear equation with variable coefficients. To attempt a solution, try a substitution inspired by reduction of order: let (y = e^{\lambda x}). Then: [ \frac{dy}{dx} ________ __________ ________ _____ ______ ____ _______ _________ ______ _______ _______ ____.
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