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Differential Equations

Question

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Solve $dy over dx space equals space left parenthesis 4 straight x plus straight y plus 1 right parenthesis squared.$

Solution

The given differential equation is

dy over dx space equals space space left parenthesis 4 straight x plus straight y plus 1 right parenthesis squared

Put

4 straight x plus straight y plus 1 space equals space straight t comma

therefore space space 4 plus dy over dx space equals space dt over dx comma space space or space space dy over dx space equals space dt over dx minus 4

therefore space space space given space equation space becomes comma space dt over dx minus 4 space equals straight t squared

or

dt over dx space equals space straight t squared plus 4

Separating the variables, we get,

fraction numerator dt over denominator straight t squared plus 4 end fraction space equals space dx

Integrating both sides,

integral fraction numerator dt over denominator straight t squared plus left parenthesis 2 right parenthesis squared end fraction space equals space integral 1. space dx

or

1 half tan to the power of negative 1 end exponent straight t over 2 space equals space straight x plus straight c space space space or space space tan to the power of negative 1 end exponent straight t over 2 space equals space 2 space left parenthesis straight x plus straight c right parenthesis comma space space or space space straight t over 2 space equals space tan space open square brackets 2 space left parenthesis straight x plus straight c right parenthesis close square brackets

therefore

straight t space equals space 2 space tan space left square bracket space 2 space left parenthesis straight x space plus straight c right parenthesis right square bracket

or

4 straight x plus straight y plus 1 space equals space 2 space tan space left square bracket 2 space left parenthesis straight x plus straight c right parenthesis right square bracket comma space which space is space the space required space solution.

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