MHT CET · Maths · Differential Equations
The integrating factor of the differential equation \(x \frac{d y}{d x}+y \log x=x^{2}\) is
- A \((\log x)^{x}\)
- B \(x^{\log x}\)
- C \((\log x)^{2}\)
- D \(x^{\log (\sqrt{x})}\)
Answer & Solution
Correct Answer
(D) \(x^{\log (\sqrt{x})}\)
Step-by-step Solution
Detailed explanation
(D)
We have \(x \frac{d y}{d x}+y \log x=x^{2}\)
\(\begin{array}{l}
\therefore \frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{\log \mathrm{x}}{\mathrm{x}}\right) \mathrm{y}=\mathrm{x} \\
\therefore \text { I.F. }=\mathrm{e}^{\int \frac{\log \mathrm{x}}{\mathrm{x}} \mathrm{dx}}=\mathrm{e}^{\frac{1}{2}(\log \mathrm{x})^{2}}
\end{array}\)
We have \(x \frac{d y}{d x}+y \log x=x^{2}\)
\(\begin{array}{l}
\therefore \frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{\log \mathrm{x}}{\mathrm{x}}\right) \mathrm{y}=\mathrm{x} \\
\therefore \text { I.F. }=\mathrm{e}^{\int \frac{\log \mathrm{x}}{\mathrm{x}} \mathrm{dx}}=\mathrm{e}^{\frac{1}{2}(\log \mathrm{x})^{2}}
\end{array}\)
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