Differential Equations

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Question

If $straight x equals straight a space sint space and space straight y space equals space straight a open parentheses cost plus log space tan straight t over 2 close parentheses space find space fraction numerator straight d squared straight y over denominator dx squared end fraction$

Solution

straight y space equals space straight a open parentheses cost plus log space tan straight t over 2 close parentheses rightwards double arrow dy over dt space equals space straight a open square brackets straight d over dt left parenthesis cost right parenthesis plus straight d over dt left parenthesis log space tan straight t over 2 right parenthesis close square brackets space equals space straight a open square brackets negative sint plus cot straight t over 2 cross times sec squared straight t over 2 cross times 1 half close square brackets equals space straight a open square brackets negative sint plus fraction numerator 1 over denominator 2 sin begin display style straight t over 2 end style cos begin display style straight t over 2 end style end fraction close square brackets equals straight a open parentheses negative sint plus 1 over sint close parentheses equals straight a open parentheses fraction numerator negative sin squared straight t plus 1 over denominator sint end fraction close parentheses equals straight a fraction numerator cos squared straight t over denominator sint end fraction

straight x equals straight a space sint dx over dt equals straight a straight d over dt left parenthesis sin space straight t right parenthesis space equals space straight a space cost therefore space dy over dx equals fraction numerator open parentheses begin display style dy over dt end style close parentheses over denominator open parentheses begin display style dx over dt end style close parentheses end fraction equals space fraction numerator open parentheses straight a begin display style fraction numerator cos squared straight t over denominator sint end fraction end style close parentheses over denominator acost end fraction space equals cost over sint equals cot space straight t fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative cosec squared straight t dt over dx equals negative cosec squared straight t cross times fraction numerator 1 over denominator straight a space cost end fraction equals negative fraction numerator 1 over denominator straight a space sin squared straight t space cost end fraction