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If y=1=x/(1!)+(x^2)/(2!)+(x^3)/(3!)++(x^...

If `y=1=x/(1!)+(x^2)/(2!)+(x^3)/(3!)++(x^n)/(n !),` show that `(dy)/(dx)-y+(x^n)/(n !)=0.`

Text Solution

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`(dy)/(dx)=0+(1)/(1!)+(1)/(2!)(2x)+(1)/(3!)(3x^(2))+...+(1)/(n!)(nx^(n-1))`
`=1+(x)/(1!)+(x^(2))/(2!)+...+(x^(n-1))/((n-1)!)`
`={1+(x)/(1!)+(x^(2))/(2!)+...+(x^(n-1))/((n-1)!)+(x^n)/(n!)}-(x^(n))/(n!)`
`=y-(x^(n))/(n!)`
`"or "(dy)/(dx)-y+(x^(n))/(n!)=0`
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