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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\frac{\text{dy}}{\text{dx}}=0,\text{y}(0)=2,\text{y}'(0)=1$Function $\text{y}=\text{e}^\text{x}+1$
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Answer

Here, $\text{y}=\text{e}^{\text{x}}+1 ....(1)$
Differentiating it with respect to $x$
$\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}$
$\frac{\text{dy}}{\text{dx}}=\text{y}-1 ...(2)$
Again, differentiating it with respect to $x,$
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\frac{\text{dy}}{\text{dx}}$
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-\frac{\text{dy}}{\text{dx}}=0$
It is given differential equation. So,
$y = e^x+ 1$ is a solution of the equation
put $x - 0$ in equation $(1)$
$\Rightarrow y = e^0+ 1 = 2$
$y(0) = 2$
put $x = 0$ in equation $(2)$
$y' = e^0= 1$
$y'(0) = 1$

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