Solve: from Mathematics Differential Equations

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Differential Equations

Short Answer Type

271.

Solve:
$dy over dx plus 2 straight y space equals space sin space 3 straight x$

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272.

Solve:
$dy over dx plus 2 straight y space equals space sin space 5 straight x$

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273.

Solve:
$dy over dx plus 2 straight y space equals space cos space 3 straight x$

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274.

Solve:
$dy over dx plus 3 straight y space equals space sin space 2 straight x$

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275.
Solve:
$dy over dx plus 3 straight y space equals space cos space 2 straight x$

The given differential equation is

dy over dx plus 3 straight y space equals space cos space 2 straight x

Comparing it with

dy over dx plus straight P space straight y space equals straight Q comma

we get,
P = 3, Q = cos 2x

integral straight P space dx space equals space integral 3 space dx space equals space 3 straight x

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 3 straight x end exponent

Solution of given differential equation is

straight y. space straight e to the power of integral straight P space dx end exponent space equals space integral Qe to the power of integral straight P space dx end exponent dx plus straight c

or

straight y space. straight e to the power of 3 straight x end exponent space equals space integral straight e to the power of 3 straight x end exponent space cos space 2 straight x space dx plus straight c

or

ye to the power of 3 straight x end exponent space equals space fraction numerator 1 over denominator square root of 9 plus 4 end root end fraction space straight e to the power of 3 straight x end exponent space cos space open parentheses 2 straight x minus tan to the power of negative 1 end exponent 2 over 3 close parentheses plus straight c

or

ye to the power of 3 straight x end exponent space equals space fraction numerator 1 over denominator square root of 13 end fraction straight e to the power of 3 straight x end exponent space cos space open parentheses 2 straight x minus tan to the power of negative 1 end exponent 2 over 3 close parentheses plus straight c

therefore

straight y space equals fraction numerator 1 over denominator square root of 13 end fraction cos space open parentheses 2 straight x minus tan to the power of negative 1 end exponent 2 over 3 close parentheses plus space straight c space straight e to the power of negative 3 straight x end exponent

is the required solution.

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276.

Solve:
$dy over dx plus straight y space equals space cos space 2 straight x$

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277.

Solve:
$dy over dx plus straight y space secx space equals space tan space straight x$

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278.

Solve:
$dy over dx plus straight y space cotx space equals space straight x squared space cotx space plus space 2 straight x$

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279. Solve the differential equation:

left parenthesis 1 minus straight x squared right parenthesis space dy over dx plus straight x space straight y space equals space straight a space straight x

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Long Answer Type

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

straight x dy over dx minus straight a space straight y space equals space straight x plus 1 comma space space space straight x greater than 0

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