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9. differential equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths9. differential equations1 Mark
MCQ
If the function $y = e^{4x} + 2e^{-x}$ is a solution of the differential equation $\frac{{\frac{{{d^3}y}}{{d{x^3}}} - 13\frac{{dy}}{{dx}}}}{y} = K$ then the value of $K$ is
  • A
    $4$
  • B
    $6$
  • C
    $9$
  • ✓
    $12$
✓

Answer

Correct option: D.
$12$
d
$y = e^{4x} + 2e^{-x} ; \,\, y_1 = 4e^{4x} - 2e^{-x} ;$
$y_2 = 16e^{4x} + 2e^{-x} ; \,\,y_3 = 64e^{4x} - 2e^{-x}$
Now, $y_3 - 13y_1 = (64e^{4x} - 2e^{-x}) - 13(4e^{4x} - 2e^{-x}) = 12e^{4x} + 24e^{-x}$
$= 12(e^{4x} + 2e^{-x}) = 12y$
$\frac{{{y_3} - 13{y_1}}}{y}  = 12$

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