For each of the differential equations in find a particular solut… - Vidyadip

Question

For each of the differential equations in find a particular solution satisfying the given condition:
$\text{cos}\bigg(\frac{\text{dy}}{\text{dx}}\bigg) = \text{a}(\text{a} \in \text{R}); \ \text{y}=1$ when $x = 0$

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Answer

Given: Differention equation $\text{cos}\bigg(\frac{\text{dy}}{\text{dx}}\bigg) = \text{a}(\text{a} \in \text{R}); \ \text{y}=1$ when $x = 0$
$\ \Rightarrow \ \frac{\text{dy}}{\text{dx}}=\text{cos}^{-1}\text{a}$
$\Rightarrow \ \text{dy}=\Big(\text{cos}^{-1}\text{a}\Big)\text{dx}$
Integrating both sides, $\int 1\ \text{dy}=\int\Big(\text{cos}^{-1}\text{a}\Big)\text{dx}$
$\Rightarrow \ \text{y}=\Big(\text{cos}^{-1}\text{a}\Big)\int1\ \text{dx}$
$\ \Rightarrow \ \text{y}=\Big(\text{cos}^{-1}\text{a}\Big)\ \text{x+c}\ .....\text{(i)}$
Now putting $y = 1$ when $x = 0$ in eq. $(i),$ we get $c = 1$
Putting $c = 1$ in eq. $(i), y = (\cos ^{-1} a) x + 1$
$\Rightarrow\frac{\text{y-1}}{\text{x}}=\cos^{-1} \text{a}$
$\Rightarrow\ \cos\frac{\text{y-1}}{\text{x}}=\text{a}$

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