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

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Solution

The given equation is
y = a sin (x + b) ...(1)
Differentiating w.r.t.x, we get,

dy over dx space equals space straight a space cos 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 negative asin left parenthesis straight x plus straight b right parenthesis

fraction numerator straight d squared straight y over denominator dx squared end fraction equals space minus straight y

open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

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.

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