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

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247.

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

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Long Answer Type

248.
Solve $dy over dx plus fraction numerator 1 over denominator straight x space logx end fraction straight y space equals space 1 over straight x.$

The given differential equation is $dy over dx plus fraction numerator 1 over denominator straight x space logx end fraction. straight y space equals space 1 over straight x$
Comparing it with $dy over dx plus Py space equals space straight Q comma space we space get comma space space straight P space equals space fraction numerator 1 over denominator straight x space logx end fraction comma space straight Q space equals 1 over straight x$
$integral straight P space dx space equals space integral fraction numerator 1 over denominator straight x space logx end fraction dx space equals space integral fraction numerator begin display style 1 over straight x end style dx over denominator log space straight x end fraction space equals space log space left parenthesis log space straight x right parenthesis$
$therefore space space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of log space left parenthesis log space straight x right parenthesis end exponent space equals space log space straight x therefore space space space space solution space of space differential space equation space is space space space space space space space space space space space space space straight y space straight e to the power of integral straight P space dx end exponent space equals space integral straight Q. space straight e to the power of integral straight P space dx end exponent space dx space plus space straight c apostrophe space space or space straight y. space log space straight x space equals space integral 1 over straight x. space log space straight x space dx space plus space straight c apostrophe$

or $straight y space log space straight x space equals space integral left parenthesis log space straight x right parenthesis. space 1 over straight x dx plus straight c apostrophe space space or space space straight y space log space straight x space equals space fraction numerator left parenthesis log space straight x right parenthesis squared over denominator 2 end fraction plus straight c apostrophe$
or $2 space straight y space log space straight x space equals space left parenthesis log space straight x right parenthesis squared plus straight c$