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Let f: R rarr R be a differentiable func...

Let `f: R rarr R` be a differentiable function with `f(0)=0.` If `y=f(x)` satisfies the differential equation `dy/dx=(2+5y)(5y-2),` then the value of `lim_(x rarr - infty) f (x)` is …....

Text Solution

Verified by Experts

The correct Answer is:
`(0.40)`
We have,
`dy/dx=(2+5y)(5y-2)`
`rArr dy/(25y^(2)-4)=dx rArr 1/25(dy/(y^(2)-4/25))=dx`
On integrating both sides, we get
`1/25 int dy/(y^(2)-(2/5)^(2))=int dx`
`rArr 1/25xx1/(2xx2/5)log abs((y-2//5)/(y+2//5))=x+C`
`rArr log abs((5y-2)/(5y+2))=20(x+C)`
`rArr abs((5y-2)/(5y+2))=Ae^(20x)[therefore e^(20c)=A]`
when `x=0 rArr y = 0, than A = 1`
`therefore abs((5y-2)/(5y+2))=e^(20x)`
`lim_(x rarr - infty) abs((5f(x)-2)/(5f(x)+2))=lim_(x rarr - infty) e^(20x)`
`rArr lim_(n rarr - infty) abs((5f(x)-2)/(5f(x)+2))=0`
`rArr lim_(n rarr - infty) 5 f(x)-2=0`
`rArr lim_(n rarr - infty) f (x) = 2/5=0.4`
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