अवकल समीकरण\(\rm 2y\frac{{d} x}{{d} y}+ x = 5y^{2}\)समाकलन कारक क्या है? (y ≠ 0)

  1. \(\rm \sqrt{y}\)
  2. y2
  3. y
  4. \(\rm \frac{1}{\sqrt{y}}\)

Answer (Detailed Solution Below)

Option 1 : \(\rm \sqrt{y}\)
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संकल्पना:

समाकलन कारक, (IF) अवकल समीकरण, \(\rm\frac{dx}{dy}+Px= Q\), के लिए जहां P और Q को y का सतत फलन दिया जाता है।

 IF = \(\rm e^{\int Pdy}\)

गणना:

दिए गए समीकरण को निम्न प्रकार सरल बनाया जा सकता है,

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

मानक समीकरण \(\rm\frac{dx}{dy}+Px= Q\)के साथ समीकरणों की तुलना करने पर, हम प्राप्त करते हैं,

P = \(\rm \frac{1}{2y}\) और 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 = \(\rm \sqrt{y}\) .  (∵ \(\rm e^{a \log x}= x^{a}\))

सही विकल्प 1 है।

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