Which of the following differential equations has y = c_1e^x + c_… - Vidyadip

Question

Which of the following differential equations has $\text{y} = \text{c}_1\text{e}^\text{x} + \text{c}_2\text{e}^{-\text{x}}$ as the general solution?

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

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Answer

The given equation is:
$\text{y} = \text{c}_1\text{e}^\text{x} + \text{c}_2\text{e}^{-\text{x}} \ \ ...(1)$
Differentiating with respect to x, we get:
$\frac{\text{dy}}{\text{dx}} = \text{c}_{1}\text{e}^\text{x}-\text{c}_{2}\text{e}^{-\text{x}}$
Again, differentiating with respect to x, we get:
$\frac{\text{d}^2\text{y}}{\text{dx}^2} = \text{c}_{1}\text{e}^\text{x}+\text{c}_2\text{e}^{-\text{x}}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2} =\text{y}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{y}=0$
This is the required differential equation of the given equation of curve.
Hence, the correct answer is B.

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