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Differential Equations

Long Answer Type

281. Find the general solution of the following differential equation:

dy over dx plus fraction numerator 4 straight x over denominator straight x squared plus 1 end fraction straight y space equals negative fraction numerator 1 over denominator left parenthesis straight x squared plus 1 right parenthesis cubed end fraction

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Short Answer Type

282.

Solve: $open parentheses straight x plus straight y close parentheses space dy over dx space equals space 1.$

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Long Answer Type

283.

Solve:
$open parentheses straight x plus 2 straight y cubed close parentheses space straight y apostrophe space equals space straight y comma space space straight y greater than 0$

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Short Answer Type

284.

Solve:
$straight y space dx space plus space left parenthesis straight x minus straight y right parenthesis squared space dy space equals space 0$

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Long Answer Type

285. Find the general solution of the differential equation y dx – (x + 2 y²) dy = 0.

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286. Solve the following differential equation:
(1 + y²)dx = (tan^{– 1} y – x) dy

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287. Solve the following differential equation:

square root of 1 minus straight y squared end root space dx equals left parenthesis sin to the power of negative 1 end exponent straight y space minus straight x right parenthesis space dy

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Short Answer Type

288. Find a one parameter family of solutions of each of the following differential equation e^–y sec² y dy = dx + x dy.

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Long Answer Type

289. Find the particular solution of the differential equation:

dy over dx plus straight y space cotx space equals space 2 straight x plus straight x squared space cotx space space space left parenthesis straight x not equal to 0 right parenthesis

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290. For the given differential equation, find a particular solution satisfying the given condition:
$dy over dx plus 2 straight y space tan space straight x space equals space sin space straight x space colon space straight y space space equals space 0 space space when space straight x space equals space straight pi over 3$

The given equation is

dy over dx plus 2 straight y space tan space straight x space equals space sin space straight x

Comparing with

dy over dx plus straight P space straight y space equals space straight Q comma space space we space get comma space space straight P space equals space 2 space tanx space comma space space straight Q space equals space sin space straight x

therefore space space space space space integral straight P space dx space equals space integral 2 space tan space straight x space dx space equals space minus 2 space log space cosx space equals space log space open parentheses cosx close parentheses squared therefore space space 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 to the power of log space left parenthesis cos space straight x right parenthesis to the power of negative 2 end exponent end exponent space equals space left parenthesis cos space straight x right parenthesis to the power of negative 2 end exponent space equals space fraction numerator 1 over denominator cos squared straight x end fraction

therefore

solution of given equation is

straight y. space fraction numerator 1 over denominator cos squared straight x end fraction space equals space integral space sinx. space space fraction numerator 1 over denominator cos squared straight x end fraction dx plus straight c

open square brackets because 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 dx plus straight c close square brackets

fraction numerator straight y over denominator cos squared straight x end fraction space equals space integral secx space tanx space dx space plus space straight c

or space space space fraction numerator straight y over denominator cos squared straight x end fraction space equals secx space plus space straight c or space space space straight y space equals space cos space straight x space plus space straight c space cos to the power of 2 straight x end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis because space space space space space space straight y space equals space 0 space space space space when space straight x space equals space straight pi over 3 therefore space space space space space space 0 space equals space 1 half plus straight c space open parentheses 1 half close parentheses squared space space space space space space space space space space space therefore space space space straight c space equals space minus 2 therefore space space space from space left parenthesis 1 right parenthesis comma space space straight y space equals space cosx minus 2 space cos squared straight x.

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