చలనసమీకరణము యొక్క అనుకలన గుణాంకము \(\rm 2y\frac{dx}{dy}+ x = 5y^{2}\)

  1. \(\rm \sqrt{y}\)
  2. y 2
  3. y
  4. \(\rm \frac{1}{\sqrt{y}}\)

Answer (Detailed Solution Below)

Option 1 : \(\rm \sqrt{y}\)
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Detailed Solution

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కాన్సెప్ట్:

అనుకలన గుణాంకము, (IF) చలనసమీకరణము, \(\rm\frac{dx}{dy}+Px= Q\) , ఇక్కడ P మరియు Q లకు y యొక్క నిరంతర ఫంక్షన్ ఇవ్వబడుతుంది.

IF = \(\rm e^{\int Pdy}\)

లెక్కింపు:

ఇచ్చిన సమీకరణాన్ని ఇలా సరళీకరించవచ్చు,

\(\rm \frac{\mathrm{d} x}{\mathrm{d} y}+ \frac{x}{2y} = \frac{5}{2}y\)

సమి ( i )ని ప్రామాణిక సమీ, \(\rm\frac{dx}{dy}+Px= Q\) తో పోల్చినప్పుడు, మనకు లభిస్తుంది,

P = \(\rm \frac{1}{2y}\) మరియు Q = \(\rm \frac{5}{2}y\)

∴ IF = \(\rm e^{\int Pdy}\) = \(\rm e^{\int \frac{1}{2y}dy}\)

⇒ IF = \(\rm e^{\frac{1}{2}\log y}\) = \(\rm e^{\log y^{\frac{1}{2}}}\)

IF = \(\rm \sqrt{y}\) . ( ∵ \(\rm e^{a \log x}= x^{a}\) )

సరైన ఎంపిక 1.

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