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.

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:
$straight x dy over dx plus straight y minus straight x plus straight x space straight y space cot space straight x space equals space 0 space space space left parenthesis straight x not equal to 0 right parenthesis$

The given differential equation is
   $straight x dy over dx plus straight y minus straight x plus straight x space straight y space cot space straight x space equals space 0$
or $straight x dy over dx plus left parenthesis 1 plus space straight x space cot space straight x right parenthesis space straight y space equals straight x$
or    $dy over dx plus open parentheses 1 over straight x plus cot space straight x close parentheses. straight y space equals space 1$
Comparing it with $dy over dx plus Py space equals straight Q comma space space we space get comma space space space straight P space equals space 1 over straight x plus cot space straight x comma space space straight Q space equals space 1$
$integral straight P space dx space equals space integral open parentheses 1 over straight x plus cot space straight x close parentheses space dx space equals space log space straight x plus space log space sinx space equals space log space left parenthesis straight x space sin space straight x right parenthesis$
$therefore space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of log space left parenthesis straight x space sinx right parenthesis end exponent space equals space space straight x space sin space straight x$

Solution of given differential equation is
$straight y space straight e to the power of integral straight P space dx end exponent space equals space integral straight Q space straight e to the power of integral straight P space dx end exponent dx plus straight c$
or $straight y. space left parenthesis straight x space sin space straight x right parenthesis space equals space integral space 1. space straight x space sinx space dx space plus straight c$
or    $straight x space straight y space sin space straight x space equals integral space straight x space sin space straight x space dx space plus space straight c$
or $straight x space straight y space sin space straight x space equals straight x. space left parenthesis negative cos space straight x right parenthesis minus space integral 1. space left parenthesis negative cos space straight x right parenthesis space dx plus space straight c$
or    $straight x space straight y space sin space straight x space equals space minus straight x space cos space straight x space space plus space integral space cosx space dx plus straight c$
or    $straight x space straight y space sin space straight x space equals space minus straight x space cos space straight x space plus space sin space straight x space plus straight c$
or    $straight y equals negative cot space straight x plus 1 over straight x plus fraction numerator straight c over denominator straight x space sin space straight x end fraction$
which is required solution.

74 Views

263.

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