Solve the differential equation:
$dy over dx space equals space straight y space sin space 2 straight x comma space space space given space that space space straight y left parenthesis 0 right parenthesis space equals space 1.$

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Solution

The given differential equation is

dy over dx space equals space straight y space sin space 2 straight x

Separating the variables, we get,

1 over straight y dy space equals space sin space 2 straight x space dx

Integrating,

integral 1 over straight y dy space equals space integral sin space 2 straight x space dx

therefore space space space log space open vertical bar straight y close vertical bar space equals space minus fraction numerator cos space 2 straight x over denominator 2 end fraction plus straight c

...(1)
Now

straight y left parenthesis 0 right parenthesis space equals space 1 space space space space space space space space space space space space rightwards double arrow space space space straight y space equals space 1 space space space space when space straight x space equals space 0

therefore space space log space open vertical bar 1 close vertical bar space equals space minus fraction numerator cos space 0 over denominator 2 end fraction plus straight c space space space space space rightwards double arrow space space 0 space equals space minus 1 half plus straight c space space space space rightwards double arrow space space space space straight c space equals space 1 half therefore space space from space left parenthesis 1 right parenthesis space space space log space open vertical bar straight y close vertical bar space equals space minus 1 half space cos space 2 straight x space plus 1 half space which space is space required space solution. space

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