Word Count : 457

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To solve this problem, we are given that one solution to the differential equation \[ 2x^{2}y'' + 3xy' - y = 0 \quad \text{for} \quad x > 0 \] is \( y_1(x) = x^{-1} \). We need to find a second linearly independent solution. ### Method: We will use the method of reduction of order to find the second solution. If \( y_1(x) = x^{-1} \) is one solution, the second solution, \( y_2(x) \), can be assumed in the form: \[ y_2(x) = v(x) y_1(x) = v(x) x^{-1} \] where __________ ___ _______ ____ ___ _______ _________.
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