PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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# Solution for If Y = a Xn + 1 + Bx−N and X 2 D 2 Y D X 2 = λ Y Then Write the Value of λ ? - PUC Karnataka Science Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

If y = a xn + 1 + bxn and $x^2 \frac{d^2 y}{d x^2} = \lambda y$  then write the value of λ ?

#### Solution

$y = a x^{n + 1} + b x^{- n}$

$\text { and }x^2 \frac{d^2 y}{d x^2} = \lambda y$

$\text { Now },$

$\frac{d y}{d x} = a\left( n + 1 \right) x^n - bn x^{- n - 1}$

$\text{ and } \frac{d^2 y}{d x^2} = an\left( n + 1 \right) x^{n - 1} - bn\left( - n - 1 \right) x^{- n - 2}$

$\text { Now,} x^2 \frac{d^2 y}{d x^2} = \lambda y \left[ \text { Given } \right]$

$\Rightarrow x^2 \left[ an\left( n + 1 \right) x^{n - 1} + bn\left( n + 1 \right) x^{- n - 2} \right] = \lambda\left( a x^{n + 1} + b x^{- n} \right)$

$\Rightarrow an\left( n + 1 \right) x^{n + 1} + bn\left( n + 1 \right) x^{- n} = \lambda\left( a x^{n + 1} + b x^{- n} \right)$

$\Rightarrow n\left( n + 1 \right)\left( a x^{n + 1} + b x^{- n} \right) = \lambda\left( a x^{n + 1} + b x^{- n} \right)$

$\Rightarrow \lambda = n\left( n + 1 \right)$

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Solution If Y = a Xn + 1 + Bx−N and X 2 D 2 Y D X 2 = λ Y Then Write the Value of λ ? Concept: Simple Problems on Applications of Derivatives.
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