Differential Equations

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Question

For the following differential equation, given below, find particular solution satisfying the given condition:
$dy over dx equals straight y space tanx space semicolon space space straight y space equals space 1 space space when space straight x space equals space 0$

Solution

The given differential equation is

dy over dx space equals space straight y space tanx

Separating the variables, we get,

1 over straight y dy space equals space tanx space dx

Integrating,

integral 1 over straight y dy space equals space integral tanx space dx

therefore space space space space log space open vertical bar straight y close vertical bar space equals space log space open vertical bar cosx close vertical bar space plus space straight c apostrophe space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis therefore space space space space log space open vertical bar straight y close vertical bar plus log space open vertical bar cosx close vertical bar space equals space straight c apostrophe therefore space space space log space open vertical bar straight y space cosx close vertical bar space equals space straight c apostrophe therefore space space space space open vertical bar straight y space cosx close vertical bar space equals space straight e to the power of straight c apostrophe end exponent therefore space space space straight y space cosx space equals space plus-or-minus straight e to the power of straight c space space space or space space straight y space cosx space equals space straight c

Now y = 1 when x = 0

therefore space space space 1 space space cos space 0 space equals space straight c space space space space rightwards double arrow space space space space 1 cross times 1 space equals space straight c space space space space space rightwards double arrow space space space straight c space equals space 1 therefore space space from space left parenthesis 1 right parenthesis comma space space space straight y space cosx space equals space 1 comma space space which space is space required space solution.