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Differential Equations

Short Answer Type

21.

Determine the order and degree of the differential equation:
$square root of straight x fraction numerator straight d to the power of 4 straight y over denominator dx to the power of 4 end fraction plus 5 open parentheses dy over dx close parentheses squared space equals 2 straight y squared minus straight x$

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Multiple Choice Questions

22. The degree of the differential equation

open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses cubed plus open parentheses dy over dx close parentheses squared plus sin open parentheses dy over dx close parentheses plus 1 space space equals space 0

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23. The degree of the differential equation

2 straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction minus 3 dy over dx plus straight y space equals space 0

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Short Answer Type

24. Verify that y = A cor x – B sin x is a solution of the differential equation

fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y space equals space 0.

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25. Verify that the function y = a cos x + b sin x, where a, b ∊ R is a solution of the differential equation

fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y space equals space 0.

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Long Answer Type

26. Verify that y - 3 cos (log x) + 4 sin (log x) is a solution of the differential equation $straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight y space equals space 0.$

$straight y space equals space 3 space cos space left parenthesis log space straight x right parenthesis space plus space 4 space sin space left parenthesis log space straight x right parenthesis$ ...(1)
$therefore space space space dy over dx equals 3 space sin space left parenthesis log space straight x right parenthesis. space space 1 over straight x plus 4 space cos space left parenthesis log space straight x right parenthesis. space 1 over straight x therefore space space space straight x dy over dx equals negative 3 space sin space left parenthesis log space straight x right parenthesis space plus space 4 space cos thin space left parenthesis log space straight x right parenthesis$
Again differentiating w.r.t.x, we get,
$straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx.1 space equals space minus 3 space cos space left parenthesis log space straight x right parenthesis. space 1 over straight x minus space 4 space sin space left parenthesis log space straight x right parenthesis. space 1 over straight x$
$therefore space space space space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx space equals space minus 3 space cos space left parenthesis log space straight x right parenthesis space minus space space 4 space sin space left parenthesis log space straight x right parenthesis therefore space space space space space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx equals space space minus straight y space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets therefore space space space space space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight y space equals space 0$
$therefore space space space space straight y space equals space 3 space cos space left parenthesis log space straight x right parenthesis space plus space 4 space sin space left parenthesis log space straight x right parenthesis$ is a solution of the differential equation
$straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight y space equals space 0.$