PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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# Solution for If Y = Ae2x + Be−X, Show That, D 2 Y D X 2 − D Y D X − 2 Y = 0 ? - PUC Karnataka Science Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

If y = ae2x + be−x, show that, $\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0$ ?

#### Solution

Here,

$y = a e^{2x} + b e^{- x}$

$\text { Differentiating w . r . t . x, we get }$

$\frac{d y}{d x} = 2a e^{2x} - b e^{- x}$

$\text { Differentiating again w . r . t . x, we get }$

$\frac{d^2 y}{d x^2} = 4a e^{2x} + b e^{- x}$

$\Rightarrow \frac{d^2 y}{d x^2} = 2a e^{2x} - b e^{- x} + 2\left( a e^{2x} + b e^{- x} \right)$

$\Rightarrow \frac{d^2 y}{d x^2} = \frac{d y}{d x} + 2y$

$\Rightarrow \frac{d^2 y}{d x^2} - \frac{d y}{d x} - 2y = 0$

Hence proved.

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Solution If Y = Ae2x + Be−X, Show That, D 2 Y D X 2 − D Y D X − 2 Y = 0 ? Concept: Simple Problems on Applications of Derivatives.
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