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

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

$xy space equals space log space straight y space plus space straight C$
Differentiate w.r.t.x, we get
$xy apostrophe plus straight y.1 space equals space space 1 over straight y. straight y apostrophe space plus space 0$
or $xy apostrophe plus straight y space equals space fraction numerator straight y apostrophe over denominator straight y end fraction$
$rightwards double arrow space space space xyy apostrophe plus straight y squared space equals space straight y apostrophe rightwards double arrow space space space space space space space space space space space straight y squared space equals space straight y apostrophe minus xyy apostrophe rightwards double arrow space space space space space space space space space straight y squared space equals space straight y apostrophe left parenthesis 1 minus xy right parenthesis rightwards double arrow space space space space space space space space straight y apostrophe space equals space fraction numerator straight y squared over denominator 1 minus xy end fraction$
$therefore space space xy space equals space logy plus straight C space is space straight a space solution space of space straight y apostrophe space equals space fraction numerator straight y squared over denominator 1 minus xy end fraction.$

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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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