Word Count : 548

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The given differential equation is: \[ 2x^2y'' + 3xy' - y = 0, \quad x > 0 \] and one solution is \( y_1(x) = x^{-1} \). We are tasked with finding a second linearly independent solution. ### Step 1: Use the method of reduction of order. If \( y_1(x) = x^{-1} \) is a solution, we assume the second solution is of the form: \[ y_2(x) = v(x) \cdot y_1(x) = v(x) \cdot x^{-1} \] where \( v(x) \) is an unknown function to be determined. ### Step 2: Compute derivatives of \( y_2(x) \). We need the first and second derivatives of \( y_2(x) \): \[ y_2(x) _______ ______ ________ ________ ________ ____ ___ _____ ________ _________ ___ _________.
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