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
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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
             

    The given differential equation is
    y = ex (a cos x + b sin x)
    ⇒    e–x y = a cos x + b sin x    ...(1)
    Differentiating (1) twice w.r.t. .x. we get
                      straight e to the power of negative straight x end exponent dy over dx plus straight y space straight e to the power of negative straight x end exponent left parenthesis negative 1 right parenthesis space equals space minus straight a space sinx space plus space straight b space cosx
    and        straight e to the power of negative straight x end exponent fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx straight e to the power of negative straight x end exponent left parenthesis negative 1 right parenthesis minus open curly brackets negative ye to the power of negative straight x end exponent plus straight e to the power of negative straight x end exponent dy over dx close curly brackets space equals space minus straight a space cosx space minus space straight b space sinx
    or            straight e to the power of negative straight x end exponent fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight e to the power of negative straight x end exponent dy over dx plus straight y space straight e to the power of negative straight x end exponent space equals space minus straight y space straight e to the power of negative straight x end exponent                     [because space of space left parenthesis 1 right parenthesis]
    or         straight e to the power of negative straight x end exponent fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight e to the power of negative straight x end exponent dy over dx plus 2 space straight y space straight e to the power of negative straight x end exponent space equals space 0 space space or space space straight e to the power of negative straight x end exponent open curly brackets fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 dy over dx plus 2 straight y close curly brackets space equals space 0
    or            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
    Hence the result. 

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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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