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Differential Equations

Short Answer Type

61. Form the differential equation representing the family of curves y = a cos (x + b) where a and b are arbitrary constants.

The given equation is $straight y space equals space straight a space cos left parenthesis straight x plus straight b right parenthesis$ ...(1)
Differentiating w.r.t.x, we get, $dy over dx equals negative straight a space sin left parenthesis straight x plus straight b right parenthesis$
Again differentiating w.r.t. x, we get,
$fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space minus straight a space cos space left parenthesis straight x plus straight b right parenthesis space space space space space space or space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative straight y space space space space space space space space space space left square bracket because space space of space left parenthesis 1 right parenthesis right square bracket$
or $fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y space equals space 0$
which is required differential equation.