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Let y(x) be a function satisfying (d^(2)...

Let `y(x)` be a function satisfying `(d^(2)y)/(dx^(2))-(dy)/(dx)+e^(2x)=0`, y(0)= and `y^(')(0)=1`. If maximum value of `y(x)` is `y(alpha)`, then integral part of `2alpha` is……………..

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The correct Answer is:
1
We have `(d^(2)y)/(dx^(2))-(dy)/(dx)+e^(2x)=0`
Put `(dy)/(dx)=t`
`therefore (dt)/(dx) -t=-e^(2x)`, which is linear differential equation
`rArr` solution is `te^(-x)=-inte^(2x)e^(-x)dx+c`
`rArr (dy)/(dx) e^(-x)=-e^(x)+c`
`rArr y^(')(0)=1 rArr c=2`
`therefore y=2e^(x)-e^(2x)/2+c^(')`
`therefore y(x) = 2e^(x)- e^(2x)/2+1/2`
`y(x) le5/2` for `x=log_(2)2`
So, `[2alpha]=[log_(e)4]=1`
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