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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

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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:
$straight y space equals straight a space straight e to the power of straight x space plus space straight b space straight e to the power of negative straight x end exponent plus straight x squared$ : $fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight y plus straight x squared minus 2 space equals space 0$

Here, $straight y space equals space ae to the power of straight x plus be to the power of negative straight x end exponent plus straight x squared$
$therefore space space space space space space dy over dx space equals space ae to the power of straight x minus be to the power of negative straight x end exponent plus 2 straight x therefore space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space ae to the power of straight x plus be to the power of negative straight x end exponent plus 2 therefore space space space straight L. straight H. straight S. space equals space fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight y plus straight x squared minus 2 space space space space space space space space space space space space space space space space space space space equals straight a space straight e to the power of straight x plus be to the power of negative straight x end exponent plus 2 minus left parenthesis ae to the power of straight x plus be to the power of negative straight x end exponent plus straight x squared right parenthesis plus straight x squared minus 2 space space space space space space space space space space space space space space space space space space space space equals space ae to the power of straight x plus be to the power of negative straight x end exponent plus 2 minus ae to the power of straight x minus be to the power of negative straight x end exponent minus straight x squared plus straight x squared minus 2 space space space space space space space space space space space space space space space space space space space space space equals space 0 space space equals space straight R. straight H. straight S. therefore space space space space space straight y space equals space straight a space straight e to the power of straight x plus be to the power of negative straight x end exponent plus straight x squared$
is a solution of the given differential equation.