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Differential Equations

Long Answer Type

251.

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

72 Views

Short Answer Type

252.

Find one-parameter families of solution curves of the following differential equations:
$straight y apostrophe plus 3 straight y space equals straight e to the power of negative 2 straight x end exponent$

71 Views

253.

Find one-parameter families of solution curves of the following differential equation:
$straight y apostrophe space minus space straight y space equals space cos space straight x$

76 Views

254.

Find one-parameter families of solution curves of the following differential equation:
y'+3 y = e^mx(m: a given real number)

83 Views

255.

Find one-parameter families of solution curves of the following differential equation:
y' - y = cos 2x

74 Views

256.

Find one-parameter families of solution curves of the following differential equation:
x y' - y = (x+1) e^-x

73 Views

257.
Find one-parameter families of solution curves of the following differential equation:
x y' + y = x⁴

The given differential equation is
x y' + y = x⁴or

dy over dx plus 1 over straight x straight y space equals space straight x cubed

Comparing it with

dy over dx plus Py space equals straight Q comma space we space get space straight P space equals 1 over straight x. straight Q space equals space straight x cubed

straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of integral 1 over straight x dx end exponent space equals space straight e to the power of log space straight x end exponent space equals space straight x

Solution of differential equation is

straight y space straight e to the power of integral space 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

straight y. space straight e space equals space integral straight x cubed. space straight x space dx plus straight c space space space space space space or space space space straight x space straight y space equals space integral straight x to the power of 4 dx plus straight c

xy space equals straight x to the power of 5 over 5 plus straight c space space space space space space space space space space space space space space space space space space space or space space space space space straight y space equals space straight x to the power of 4 over 5 plus straight c over straight x

76 Views

258.

Find one-parameter families of solution curves of the following differential equation:
x y' log x + y = log x

73 Views

259.

Find one-parameter families of solution curves of the following differential equation:
$dy over dx minus fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction comma space space straight y space equals space straight x squared plus 2$

81 Views

Long Answer Type

260.

Find one-parameter families of solution curves of the following differential equation:
$straight y apostrophe plus straight y space cosx space equals space straight e to the power of sin space straight x end exponent space cos space straight x$