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Differential Equations

Short Answer Type

241. Solve the differential equation:

straight x dy over dx plus 2 straight y space equals straight x squared.

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242.
Solve:
$straight v plus 2 straight y space equals 4 straight x$

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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243.

Solve:
$dy over dx plus straight y over straight x equals straight x squared.$

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244.

Solve:
$dy over dx plus 4 straight y space equals space 5 straight x$

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245.

Solve:
$dy over dx plus straight y space equals space 1$

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246.

Solve:
$fraction numerator dy apostrophe over denominator dx end fraction minus straight y over straight x equals space 2 straight x squared$

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247.

Solve:
$2 straight x dy over dx plus straight y space equals space 6 straight x cubed space space or space space space dy over dx plus fraction numerator straight y over denominator 2 straight x end fraction equals 3 straight x squared.$