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Differential Equations

Question

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Solve the following differential equation:
$open parentheses straight x plus straight y plus 2 close parentheses space dy over dx space equals 2$

Solution

The given differential equation is

open parentheses straight x plus straight y plus 2 close parentheses space dy over dx space equals 2

...(1)
Put x+y+2 = t,

therefore space space space space space 1 plus dy over dx equals dt over dx space space or space space dy over dx equals dt over dx minus 1 therefore space space from space left parenthesis 1 right parenthesis comma space space straight t open parentheses dt over dx minus 1 close parentheses space equals space 2 space space or space space straight t dt over dx space equals space 2 plus straight t space space or space space fraction numerator straight t over denominator 2 plus straight t end fraction dt space equals space dx

or

integral fraction numerator straight t over denominator 2 plus straight t end fraction dt space equals space integral dx

or space space integral open parentheses 1 minus fraction numerator 2 over denominator 2 plus straight t end fraction close parentheses space dt space equals space straight x space space or space space straight t minus 2 space log space open vertical bar 2 plus straight t close vertical bar space equals space straight x plus straight c or space space straight x plus straight y plus 2 minus 2 space log space space open vertical bar straight x plus straight y plus 4 close vertical bar space equals space straight x plus straight c space or space space straight y minus 2 space log space open vertical bar straight x plus straight y plus 4 close vertical bar space equals space straight c minus 2 space equals space straight C comma space say comma space is space the space required space solution.

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