Solve the following differential equation:
x cos y dy = (x e^x log x + e^x) dx.

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Solution

The given differential equation is

straight x space cos space straight y space dy space equals space left parenthesis straight x space straight e to the power of straight x space log space straight x space plus straight e to the power of straight x right parenthesis space dx

cos space straight y space dy space equals space open parentheses straight e to the power of straight x logx plus straight e to the power of straight x over straight x close parentheses dx

Integrating,

integral space cos space straight y space dy space equals space integral space straight e to the power of straight x open square brackets log space straight x space plus 1 over straight x close square brackets space dx

therefore

space sin space straight y space equals space straight e to the power of straight x logx space plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open curly brackets because space space space integral straight e to the power of straight x left square bracket straight f left parenthesis straight x right parenthesis plus straight f apostrophe left parenthesis straight x right parenthesis dx space equals space straight e to the power of straight x space straight f left parenthesis straight x right parenthesis close curly brackets

which is the required solution.

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