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

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

The given differential equation is $dy over dx plus straight y space equals space 1$
Comparing it with $dy over dx plus space straight P space straight y space equals space straight Q comma space we space get comma space space space straight P space equals space 1 comma space space straight Q space equals space 1$
$therefore space space space space integral straight P space dx space equals space integral 1. space dx space equals space straight x therefore space space space straight I. space straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of straight x therefore space space space solution space of space differential space equation comma space is space straight y. space straight e to the power of straight x space equals space integral 1. space straight e to the power of straight x space dx space plus straight c or space space space straight y space straight e to the power of straight x space equals space straight e to the power of straight x plus straight c space space space or space space space straight y space equals space 1 plus straight c space straight e to the power of negative straight x end exponent$

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