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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations1 Mark
MCQ
Choose the correct answer from the given four option. $\text{y}= \text{a}\text{e}^{\text{mx}}+\text{b}\text{e}^{-\text {mx}}$ satisfies which of the following differential equation?
  • A
    $\frac{\text{dy}}{\text{dx}}+\text {my}=0$
  • B
    $\frac{\text{dy}}{\text{dx}}-\text {my}=0$
  • ✓
    $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{m} ^2\text{y}=0$
  • D
    $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{m} ^2\text{y}=0$
✓

Answer

Correct option: C.
$\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{m} ^2\text{y}=0$
Given that, $\text{y}=\text{a}\text{e}^{\text{mx}}+\text{b}\text{e}^{- \text{mx}}$
On differentiating both sides $\text{w.r.t. x},$ we get
$\frac{\text{dy}}{\text{dx}}=\text {ma}\text{e}^{\text{mx}}+\text{bm}\text{e}^{-\text {mx}}$
Again, differentiating both sides $\text{w.r.t. x},$ we get
$\frac{\text{d}^2\text{y}}{\text {d}\text{x}^2}=\text{m}^2\text{a}\text{e}^{\text{mx}}+ \text{b}\text{m}^2\text{e}^{-\text{mx}}$
$\Rightarrow\frac{\text{d}^2\text {y}}{\text{d}\text{x}^2}=\text{m}^2(\text{a}\text{e}^{\text {mx}}+\text{b}\text{e}^{-\text{mx}})$
$\Rightarrow\frac{\text{d}^2\text {y}}{\text{d}\text{x}^2}=\text{m}^2\text{y}$
$\Rightarrow\frac{\text{d}^2\text {y}}{\text{d}\text{x}^2}-\text{m}^2\text{y}=0$

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