Find a one parameter family of solutions of each of the followin

Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Advertisement

Differential Equations

Long Answer Type

261.

Find one-parameter families of solution curves of the following differential equation:
$left parenthesis 1 plus straight x squared right parenthesis space dy space plus space 2 xy space dx space equals space cot space straight x space dx space space space space left parenthesis straight x not equal to 0 right parenthesis$

75 Views

262.

Find one-parameter families of solution curves of the following differential equation:
$straight x dy over dx plus straight y minus straight x plus straight x space straight y space cot space straight x space equals space 0 space space space left parenthesis straight x not equal to 0 right parenthesis$

74 Views

263.

Solve: $open parentheses fraction numerator straight e to the power of negative 2 square root of straight x end exponent over denominator square root of straight x end fraction minus fraction numerator straight y over denominator square root of straight x end fraction close parentheses dx over dy equals 1 comma space space space straight x not equal to space 0.$

75 Views

264. Solve the following differential equation:

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

79 Views

Advertisement

265. For the differential equation, find the general solution:

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

74 Views

266. Solve the following differential equation:

sin space straight x space dy over dx plus straight y space cosx space equals space cos space straight x space sin squared straight x

73 Views

Short Answer Type

Advertisement

267. Find a one parameter family of solutions of each of the following differential equation:
$straight y apostrophe cos squared straight x space equals space tanx minus space straight y$

The given differential equation is

dy over dx. space cos squared straight x space equals space tanx minus straight y space space space space space space space or space space space dy over dx equals tan space straight x space sec squared straight x space minus space straight y space sec squared straight x

or

dy over dx plus left parenthesis sec squared straight x right parenthesis. space straight y space equals space tan space straight x space sec squared straight x

Comparing it with

dy over dx plus straight P space straight y space equals space straight Q comma space space we space get comma space straight P space equals space sec squared straight x comma space space straight Q space equals tan space straight x space sec squared straight x

integral straight P space dx space equals space integral sec squared straight x space dx space equals space tan space straight x comma space space space space space 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 tanx

therefore

solution of differential equation is

straight y space straight e to the power of integral Pdx 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

or

straight y space straight e to the power of tanx space equals space integral tanx space sec squared straight x space. space straight e to the power of tanx dx plus straight c

...(1)
Let

straight I space equals space integral tanx space sec squared straight x. space space straight e to the power of tanx dx

Put tanx = t,

sec squared dx space equals space dt

therefore space space space space space space space space space space space space space space straight I space equals space integral straight t. space straight e to the power of straight t space dt space equals space straight t space straight e to the power of straight t space minus space integral 1. space straight e to the power of straight t space dt space equals space straight t space straight e to the power of straight t space minus space straight e to the power of straight t space equals space left parenthesis straight t minus 1 right parenthesis space straight e to the power of straight t space equals space left parenthesis tanx space minus space 1 right parenthesis straight e to the power of tanx

therefore space space space space from space left parenthesis 1 right parenthesis comma space space space straight y space straight e to the power of tanx space equals space left parenthesis tanx minus 1 right parenthesis space straight e to the power of tanx plus straight c

or

straight y equals tan space straight x minus 1 space plus space straight c space straight e to the power of negative tanx end exponent

which is required solution.

74 Views

Advertisement

268. Find a one parameter family of solutions of each of the following differential equation:

dy over dx equals negative fraction numerator straight x plus straight y space cosx over denominator 1 plus sinx end fraction

76 Views

Advertisement

Long Answer Type

269.

Solve:
$straight y apostrophe minus 2 straight y space equals space cos space 3 straight x$

71 Views

Short Answer Type

270.

Solve:
$straight y apostrophe plus straight y space equals space sin space straight x$

85 Views

Advertisement