Differential Equations

Question

Solve:
$straight v plus 2 straight y space equals 4 straight x$

Solution

The given differential equation is $dy over dx plus 2 straight y space equals space 4 straight x$
Comparing it with $dy over dx plus Py space equals space straight Q comma space we space get.$
P = 2, Q = 4 x
$therefore space space space integral straight P space dx space equals space integral 2 space dx space equals space 2 space straight x therefore space space space straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e squared therefore space space space solution space of space differential space equation space is$
   $straight y. space straight e to the power of 2 straight x end exponent space space space integral space 4 straight x. space straight e to the power of 2 straight x end exponent space dx plus straight c space space or space space space straight y space straight e to the power of 2 straight x end exponent space equals space 4 space integral straight x space straight e to the power of 2 straight x end exponent space dx$
or    $straight y. straight e to the power of 2 straight x end exponent space equals space 4 open square brackets straight x straight e to the power of 2 straight x end exponent over 2 minus integral 1. space straight e to the power of 2 straight x end exponent over 2 dx close square brackets plus straight c$
or $straight y. straight e to the power of 2 straight x end exponent space equals space 2 space straight x space straight e to the power of 2 straight x end exponent minus 2 integral straight e to the power of 2 straight x end exponent dx plus straight c space space or space space straight y space straight e to the power of 2 straight x end exponent space equals space 2 straight x space straight e to the power of 2 straight x end exponent minus 2. straight e to the power of 2 straight x end exponent over 2 plus straight c$
or    $straight y. space straight e to the power of 2 straight x end exponent space equals space 2 space straight x space straight e to the power of 2 straight x end exponent space space space space comma space space space straight e to the power of 2 straight x end exponent plus straight c space space space or space space straight y space equals space 2 straight x minus 1 space plus space straight c space straight e to the power of 2 straight x end exponent$

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Differential Equations

Solve:
$straight v plus 2 straight y space equals 4 straight x$

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