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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:
$dy over dx plus 4 straight y space equals space 5 straight x$

The given differential equation is

dy over dx plus 4 straight y space equals space 5 straight x

Comparing it with

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

we get,
P = 4, Q = 5x

integral straight P space dx space equals space integral 4 space dx space equals space 4 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 4 straight x end exponent

Solution of given differential equation is

ye to the power of integral straight P space dx end exponent space equals space integral space straight Q space straight e to the power of integral space straight P space dx end exponent dx space plus space straight c

ye to the power of 4 straight x end exponent space equals space integral 5 space straight x space straight e to the power of 4 straight x end exponent dx space plus straight c

or space space space space space straight y space straight e to the power of 4 straight x end exponent space equals space 5 space integral straight x space straight e to the power of 4 straight x end exponent dx plus straight c or space space space space straight y space straight e to the power of 4 straight x end exponent space equals space 5 space open square brackets straight x space straight e to the power of 4 straight x end exponent over 4 minus space integral 1. space straight e to the power of 4 straight x end exponent over 4 dx close square brackets plus straight c or space space space space straight y space straight e to the power of 4 straight x end exponent space equals space 5 over 4 straight x space straight e to the power of 4 straight x end exponent minus 5 over 4 integral straight e to the power of 4 straight x end exponent dx space plus space straight c or space space space straight y space straight e to the power of 4 straight x end exponent space equals space 5 over 4 straight x space straight e to the power of 4 straight x end exponent space minus space 5 over 4. space straight e to the power of 4 straight x end exponent over 4 plus straight c therefore space space space space space space space space straight y space equals space 5 over 4 straight x minus 5 over 16 plus ce to the power of negative 4 straight x end exponent

is required solution.

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

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Short Answer Type

249.

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

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250.

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

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Previous Year Papers

Test Series

Test Yourself

Short Answer Type

244. Solve:

Long Answer Type

Short Answer Type

244.
Solve:
$dy over dx plus 4 straight y space equals space 5 straight x$