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A differential equation associated to the primitively $y = a + b e^{5 x} + c e^{- 7 x}$ , is

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$y_{3} + 2 y_{2} - y_{1} = 0$

$4 y_{3} + 5 y_{2} - 20 y_{1} = 0$

$y_{3} + 2 y_{2} - 35 y_{1} = 0$

none of these

answer is C.

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Detailed Solution

Differentiating the given equation successively, we get

$\begin{aligned} y_{1} = 5 b e^{5 x} - 7 c e^{- 7 x} \\ y_{2} = 25 b e^{5 x} + 49 c e^{- 7 x} \end{aligned}$

and $y / 3 = 125 b e^{5 x} - 343 c e^{- 7 x}$

Multiplying (i) by 7 and then adding to (ii), we get

$y_{2} + 7 y_{1} = 60 b e^{5 x}$

Multiplying (i) by 5 and then subtracting it from (ii), we get

$y_{2} - 5 y_{1} = 84 c c^{- 7 x}$

Putting the values of band c, obtained from (iv) and (v) respectively, in (i), we get

$y_{3} + 2 y_{2} - 35 y_{1} = 0$

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