Solve the following differential equations: dy dx =y x, y(0)=1 - Vidyadip

Question

Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{ x, y}(0)=1$

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Answer

$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{ x, y}(0)=1$
$\Rightarrow\frac{1}{\text{y}}\text{dy}=\tan\text{ x dx}$
Integrating both sides, we get
$\int \frac{1}{\text{y}}\text{dy}=\int\tan\text{ x dx}$
$\Rightarrow\log|\text{y}|=\log|\sec\text{x}|+\text{C}...(1)$
We know that at $\text{x}=0$ and $\text{y}=1.$
Substituting the values of x and y in (1), we get
$\log|1|=\log|1|+\text{C}$
$\Rightarrow\text{C}=0$
substituting the value of C in (1), we get
$\log|\text{y}|=\log|\sec\text{x}|+0$
$\Rightarrow\text{y}=\sec\text{x}$
Hence, $\text{y}=\sec\text{x},$ where $\text{x}\in\Big(\frac{-\pi}{2},\frac{\pi}{2}\Big),$ is the required solution.

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