The differential equation by eliminating the arbithary constant from the equation $x y = a e^{x} + b e^{- x} + x^{2}$ is

x y_{2} + 2 y_{1} - x y = 2 - x^{2}

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x y_{2} + 2 y_{1} - x y = 2

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x y_{2} - 2 y_{1} - x y = 2

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x y_{2} - 2 y_{1} + x y = 2

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Solution

The correct option is A $x y_{2} + 2 y_{1} - x y = 2 - x^{2}$
The given equation is $x y = a e^{x} + b e^{- x} + x^{2}$ .......... (1)

There are two arbitrary constants $a$ and $b$ , so differentiate the equation w.r.t. $x$ two times,

Differentiating (1) w.r.t $x$ we get

$x y_{1} + y = a e^{x} - b e^{- x} + 2 x$ .......... (2)

Differentiating (2) w.r.t $x$ we get

$x y_{2} + y_{1} + y_{1} = a e^{x} + b e^{- x} + 2$

$x y_{2} + 2 y_{1} = a e^{x} + b e^{- x} + 2$ .......... (3)

Now, eliminate the arbitrary constants using these three equations.

From (1) and (3),

$x y_{2} + 2 y_{1} = x y - x^{2} + 2$

$x y_{2} + 2 y_{1} - x y = 2 - x^{2}$

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