For problem given below, verify that the given function (implici
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    Differential Equations

    Multiple Choice QuestionsShort 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.

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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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