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Question

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Find the general solution of each of the following equations:
$left parenthesis straight x plus 2 right parenthesis space dy over dx space equals space straight x squared plus 4 straight x minus 9$

Solution

The given differential equation is

left parenthesis straight x plus 2 right parenthesis space dy over dx space equals space straight x squared plus 4 straight x minus 9

or

dy over dx space equals space fraction numerator straight x squared plus 4 straight x minus 9 over denominator straight x plus 2 end fraction

Separating the variables, we get

dy space equals space open parentheses fraction numerator straight x squared plus 4 straight x minus 9 over denominator straight x plus 2 end fraction close parentheses space dx

or

dy space equals open parentheses fraction numerator straight x squared plus 4 straight x minus 9 over denominator straight x plus 2 end fraction close parentheses dx

or

dy space equals space open parentheses straight x plus 2 minus fraction numerator 13 over denominator straight x plus 2 end fraction close parentheses dx

Integrating,

straight y equals straight x squared over 2 plus 2 straight x minus 13 space log space open vertical bar straight x plus 2 close vertical bar space plus space straight c

which is required solution.

straight x plus 2 space space fraction numerator straight x plus 2 over denominator long division enclose straight x squared plus 4 straight x minus 9 end enclose end fraction space space space space space space space space space space space space space space space space straight x squared plus 2 straight x space space space space space space space space space space space space space space space space minus space space minus 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 2 straight x space space minus space 9 space space space space space space space space space space space space space 2 straight x space plus space 4 space space space space space space space space space space space space space space minus 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 space space space space space space space space 13

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