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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{y}=0,\text{y}(0)=2,\text{y}'(0)=0$Function $\text{y}=\text{e}^\text{x}+\text{e}^{-\text{x}}$
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Answer

We have $\text{y}=\text{e}^{\text{x}}+\text{e}^{\text{-x}} ...(1)$ Differentiating both sides of (1) with respect to $x,$ we get $\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}-\text{e}^{\text{x}} ...(2)$ Differentiating both sides of $(2)$ with respect to $x,$ we get $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\text{e}^{\text{x}}+\text{e}^{\text{-x}}$
$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\text{y}$ [Using (1)]$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-\text{y}=0$
It is the given differential equation.Therefore, $y = e^x+ e^{-x} $ satisfies the given differential equation.
Also, when $x = 0; = e^0+ e^0= 1 + 1, i.e. y(0) = 2.$
And, when $x = 0; y_1= e^0- e^0= 1 - 1, i.e. y'(0) = 0$
 Hence, $y = e^x+ e^{-x}$​​​​​​​ ^is the solution to the given initial value problem.

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