Question

for given differential equation find a particular solution. cos(dy/dx) = a (a belong R); y = 1 when x = 0

Answer 1

$\underline{ \large{ \text{Answer}}} :$

$\text{Given that,} \\ \\ \text{cos} (\frac{ \text{dy}}{ \text{dx}} ) = \text{a} \\ \\ \to \frac{ \text{dy}}{ \text{dx}} = { \text{cos}}^{ - 1} \text{a} \\ \\ \to \text{dy} = { \text{cos}}^{ - 1} \text{a} \: \: \text{dx}$

$\text{Now, on integration, we get -} \\ \\ \int \text{dy} = { \text{cos}}^{ - 1} \text{a} \: \: \int \text{dx} \\ \\ \to \text{y} = ({ \text{cos}}^{ - 1} \text{a} ) \: \text{x} + \text{c}, \: \text{where c = integral constant}$

$\text{When x = 0 and y = 1, we have} \\ \\ 1 = ( { \text{cos}}^{ - 1} \text{a}) \times 0 + \text{c} \\ \\ \to \text{c = 1} \\ \\ \therefore \text{The required solution be} \\ \\ \boxed{ \bold{ \text{y} = ({ \text{cos}}^{ - 1} \text{a}) \: \text{x} + 1 }}$

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