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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 options.The differential equation for which $\text{y}=\text{a}\cos\text{x}+\text{b}\sin\text{x}$ is a solution, is:
  • ✓
    $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+\text{y}=0$
  • B
    $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}-\text{y}=0$
  • C
    $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+(\text{a}+\text{b})\text{y}=0$
  • D
    $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+(\text{a}-\text{b})\text{y}=0$
✓

Answer

Correct option: A.
$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+\text{y}=0$
Given equation is, $\text{y}=\text{a}\cos\text{x}+\text{b}\sin\text{x}$

On differentiating both sides w.r.t.x. we get

$\frac{\text{dy}}{\text{dx}}=-\text{a}\sin\text{x}+\text{b}\cos\text{dx}$

Again, differentiating w.r.t.x. we get

$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}=-\text{a}\sin\text{x}+\text{b}\cos\text{dx}$

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

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

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