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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equations:
$\text{x}\frac{\text{dy}}{\text{dx}}+\cot\text{y}=0,$ given that $\text{y}=\frac{\pi}{4},$ when $\text{x}=\sqrt{2}.$
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Answer

We have,
$\text{x}\frac{\text{dy}}{\text{dx}}+\cot\text{y}=0$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}=-\cot\text{y}$
$\Rightarrow\tan\text{y dy}=-\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\tan\text{y dy}=-\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\log|\sec\text{y}|=-\log|\text{x}|+\log\text{C}$
$\Rightarrow\log(|\text{x}||\sec\text{y}|)=\log\text{C}$
$\Rightarrow\text{x}\sec\text{y = C}...(1)$
Given: $\text{x}=\sqrt{2},\text{y}=\frac{\pi}{4}.$
Substituting the values of x and y in (1), we get
$\sqrt{2}\sec\frac{\pi}{4}=\text{C}$
$\Rightarrow\text{C}=2$
Substituting the value of C in (1), we get
$\text{x}\sec\text{y}=2$
$\Rightarrow\text{x}=2\cos\text{y}$
Hence, $\text{x}=2\cos\text{y}$ is the required solution.

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