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

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246.
Solve:
$fraction numerator dy apostrophe over denominator dx end fraction minus straight y over straight x equals space 2 straight x squared$

The given differential equation is $dy over dx minus straight y over straight x space equals space 2 straight x squared$
Comparing it with $dy over dx plus straight P space straight y space equals space straight Q comma space we space get comma space straight P space equals negative 1 over straight x comma space space straight Q space equals space 2 straight x squared.$

$integral straight P space dx space equals space minus integral 1 over straight x dx space equals space minus log space straight x space equals space log space straight x to the power of negative 1 end exponent$
$therefore space space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of logx to the power of negative 1 end exponent end exponent space equals space straight x to the power of negative 1 end exponent space equals space 1 over straight x$
$therefore$ solution of differential equation is,
$straight y space. straight e to the power of integral straight P space dx end exponent space equals space integral straight Q. space straight e to the power of integral straight P space dx end exponent dx space plus space straight c$
or $straight y.1 over straight x space equals space integral 2 space straight x squared. space 1 over straight x dx plus straight c space space space space or space space space straight y over straight x space equals space 2 integral straight x space dx plus straight c space space or space space straight y over straight x equals straight x squared plus straight c space space or space space straight y space equals space straight x cubed plus cx$

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247.

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