In the following, verify that the given functions, (explicit or

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Differential Equations

Short Answer Type

31. In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:
y = cos x + C : y' + sinx = 0

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32. In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:
$straight y space equals space square root of 1 plus straight x squared end root$ : $straight y apostrophe space equals space fraction numerator xy over denominator 1 plus straight x squared end fraction$

$straight y space equals space square root of 1 plus straight x squared end root$ ...(1)
$therefore$    $straight y apostrophe space equals space fraction numerator 2 straight x over denominator 2 square root of 1 plus straight y squared end root end fraction space space or space space space straight y apostrophe space equals space fraction numerator straight x over denominator square root of 1 plus straight x squared end root end fraction$ ...(2)
R.H.S. = $fraction numerator xy over denominator 1 plus straight x squared end fraction space equals space fraction numerator straight x square root of 1 plus straight x squared end root over denominator 1 plus straight x squared end fraction$    $open square brackets because space space of space space left parenthesis 1 right parenthesis close square brackets$

$equals space fraction numerator straight x over denominator square root of 1 plus straight x squared end root end fraction space equals straight y apostrophe$    $open square brackets because space of space left parenthesis 2 right parenthesis close square brackets$
$equals space straight L. straight H. straight S.$
$therefore space space space straight y space equals space square root of 1 plus straight x squared end root$ is a solution of $straight y apostrophe space equals space fraction numerator xy over denominator 1 plus straight x squared end fraction$

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33. In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:

straight y space equals space straight x space sinx

:

xy apostrophe space equals space straight y plus straight x square root of straight x squared minus straight y squared end root

left parenthesis straight x not equal to 0 space and space straight x greater than straight y space or space straight x less than negative straight y right parenthesis

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34. In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:

xy space equals space log space straight y space plus space straight C

colon space space straight y apostrophe space equals space fraction numerator straight y squared over denominator 1 minus straight x space straight y end fraction left parenthesis space xy not equal to 1 right parenthesis

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35. In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:

straight y minus cos space straight y space equals space straight x

:

left parenthesis straight y space sin space straight y space plus space cos space straight y space plus space straight x right parenthesis space straight y apostrophe space equals space straight y

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36.

In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:

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37. In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:

straight y space equals space square root of straight a squared minus straight x squared end root space comma space straight x space element of space left parenthesis negative straight a comma space straight a right parenthesis space colon space space space space space space straight x plus straight y space dy over dx space equals space 0 space space space left parenthesis straight y not equal to 0 right parenthesis

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38. For problem given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:

straight y space equals straight a space straight e to the power of straight x space plus space straight b space straight e to the power of negative straight x end exponent plus straight x squared

:

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

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39. For problem given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:

straight y equals straight e to the power of straight x left parenthesis straight a space cosx space plus space straight b space sin space straight x right parenthesis space colon thin space fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 dy over dx plus 2 straight y space equals space 0

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40. For problem given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:

straight y space equals space straight x space sin space 3 straight x

:

fraction numerator straight d squared straight y over denominator dx squared end fraction plus 9 straight y minus 6 space cos space 3 straight x space equals space 0

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