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Differential Equations

Question

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Solve:
$dy over dx space equals space tan space left parenthesis straight x plus straight y right parenthesis$

Solution

The given differential equation is

dy over dx space equals space tan left parenthesis straight x plus straight y right parenthesis

Put

straight x plus straight y space equals straight t comma space space space space space space space space space space space space space space space space space therefore space space space 1 plus dy over dx space equals dt over dx space space space or space space space space dy over dx equals space dt over dx minus 1

therefore space space space space from space left parenthesis 1 right parenthesis comma space space space dt over dx minus 1 space equals space tan space straight t space space space space space or space space space space dt over dx space equals space 1 plus tan space straight t

Separating the variables, we get,

fraction numerator 1 over denominator 1 plus tan space straight t end fraction dt space equals dx space space space or space space space fraction numerator 1 over denominator 1 plus begin display style fraction numerator sin space straight t over denominator cos space straight t end fraction end style end fraction dt space equals space dx

or

fraction numerator cos space straight t over denominator sin space straight t plus cos space straight t end fraction dt space equals space dx space space space space space rightwards double arrow space space space space integral fraction numerator cos space straight t over denominator sin space straight t space plus space cos space straight t end fraction dt space equals space integral 1 space dx

rightwards double arrow space space space space 1 half space integral fraction numerator left parenthesis sint space plus space cost right parenthesis space plus space left parenthesis cost space minus space sint right parenthesis over denominator sint plus cost end fraction dt space equals space integral 1 space dx or space space space 1 half integral open parentheses 1 plus fraction numerator cost space minus sint over denominator sint plus cost end fraction close parentheses dt space equals space integral 1 space dx or space space space 1 half integral open square brackets straight t space plus space log space open vertical bar sin space straight t space plus space cos space straight t close vertical bar close square brackets space equals space straight x plus straight c or space space space 1 half open square brackets straight x plus straight y plus log space open vertical bar sin space left parenthesis straight x plus straight y right parenthesis plus cos space left parenthesis straight x plus straight y right parenthesis close vertical bar close square brackets space equals space straight x plus straight c

which is required solution.

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