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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS1 Mark
Question
Which of the following differentials equation has $\text{y}=\text{C}_{1}\text{e}^{\text{x}}+\text{C}_{2}\text{e}^{-\text{x}}$ as the general solution?
  1. $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{y}=0$
  2. $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-\text{y}=0$
  3. $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{1}=0$
  4. $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-\text{1}=0$ 
✓

Answer

  1. $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-1=0$

Solution:

We have,

$\text{y}=\text{C}_{1}\text{e}^{\text{x}}+\text{C}_{2}\text{e}^{-\text{x}}\ ...(\text{i})$

Differentiating both sides of (i) with we get,

$\frac{\text{dy}}{\text{dx}}=\text{C}_{1}\text{e}^{\text{x}}+\text{C}_{2}\text{e}^{-\text{x}}\ ...(\text{ii})$

Differentiating both sides of (ii) with we get,

$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\text{C}_{1}\text{e}^{\text{x}}+\text{C}_{2}\text{e}^{-\text{x}}$

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

$\Rightarrow \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-\text{y}=0$

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