InterviewSolution
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InterviewSolution
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If y= y(x) is the solution of the differential equation `dy/dx=(tan x-y) sec^(2)x, x in(-pi/2,pi/2),` such that y (0)=0, than `y(-pi/4)` is equal to(a)` 1/e-2 `(b) `1/2-e `(c) `2+1/e`(d)` e-2`A. (a)` 1/e-2 `B. (b) `1/2-e `C. (c) `2+1/e`D. (d)` e-2` |
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Answer» Correct Answer - (d) Given differential equation ` dy/dx=(tan x-y)sec^(2)x` `rArr dy/dx +(sec^(2) x)y=sec^(2)x tan x,` which is linear differential equation of the form `dy/dx+Py=Q,` where `P=sec^(2) x and Q=sec^(2) x tan x` `IF=e^(int sec^(2) x dx)=e^(tan x)` So, solution of given dofferential equation is `yxxIF= int(QxxIF)d+c` `y(e^tanx)=int e^(tanx). sec^(2)xtan x dx +C` Let `tan x=t rArr sec^(2) xdx =dt` `ye^tanx =int e^(t) cdot t dt + C=te^(t)-int e ^(t) dt +C` [using integration by parts method] `= e^(t) (t-1)+C` `rArr y cdot e^tan x=e^(tan x ) (tan x-1)+C [therefore t =tan x]` `therefore y(0)=0` ` rArr 0=1(0-1)+C rArr C=1` `therefore y cdot e^(tan x)=e^(tan x)(tan x-1)+1` Now, at `x=-pi/4` `ye^(-2) =e^(-1)(-1-1)+1` ` rArr ye^(-1) =2e^(-1)+1 rArr y=e-2` |
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