InterviewSolution
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InterviewSolution
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Solve the differential equation ` (cos ^(2) x ) (dy)/(dx) + y = tan x ( 0 le x lt (pi)/(2))` |
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Answer» The given differential equation may be written as ` (dy)/(dx) + (sec ^(2) x )y = (sec ^(2) x ) tan x " " ` ... (i) This is of th form ` (dy)/(dx) + Py = Q`, where `P= sec ^(2) x and Q = sec ^(2) x tanx `. Thus, the given differential equation is linear. `IF = e ^(int Pdx) = e ^(int sec ^(2)x dx ) = e ^(tanx )` So, its solution is given by `y xx IF = int ( Q xx IF )dx + C ` i.e, `yxx e ^(tanx ) = int e ^(tanx ) (sec ^(2) x ) tan x dx + C ` `" " = int underset("I")t underset ("II") (e^(t)) dt, ` where ` tanx = t and sec ^(2) x d x = dt ` `" " = t e ^(t) - int 1* e ^(t) dt + C ` ` " " = t e ^(t) - e ^(t) +C = e^(t) (t -1 ) +C ` `" "= e ^(tan x ) (tanx - 1 ) +C `. Hence, `y = (tanx - 1 ) + Ce ^(-tan x )` is the required solution. |
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