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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS1 Mark
Question
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:

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

  2. $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}-\text{y}=0$

  3. $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+(\text{a}+\text{b})\text{y}=0$

  4. $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+(\text{a}-\text{b})\text{y}=0$

✓

Answer

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

Solutuion:

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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