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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

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

$straight y space equals space square root of straight a squared minus straight x squared end root$ ...(1)
$therefore space$ $straight y apostrophe space equals space fraction numerator negative 2 straight x over denominator 2 square root of straight a squared minus straight x squared end root end fraction$
$rightwards double arrow space space space space straight y apostrophe equals negative fraction numerator straight x over denominator square root of straight a squared minus straight x squared end root end fraction$
$rightwards double arrow space space space space space space space space space dy over dx equals negative straight x over straight y space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets rightwards double arrow space space space space space space straight y dy over dx space equals space minus straight x rightwards double arrow space space space space straight x plus straight y dy over dx space equals space 0 space therefore space space space straight y space equals space square root of straight a squared minus straight x squared end root space is space straight a space solution space of space straight x plus straight y dy over dx space equals space 0.$