Question:medium

If \(y_{1}=x^{2}\) is a known solution of \[ x^{2}y^{\prime\prime}-3xy^{\prime}+4y=0, \] then the second independent solution \(y_{2}\) is

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For a second-order linear homogeneous differential equation, \[ y''+P(x)y'+Q(x)y=0, \] if one solution \(y_{1}\) is known, immediately use \[ y_{2}=y_{1}\int \frac{e^{-\int P(x)\,dx}} {(y_{1})^{2}}\,dx. \] This formula is frequently used in university examinations and competitive exams for finding the second independent solution without solving the entire differential equation from scratch.

Updated On: Jun 29, 2026

\(\dfrac{1}{x}\)
\(x^{2}\)
\(\log x\)
\(x^{2}\log x\)

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The Correct Option is D

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