- #1
chwala
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- Homework Statement
- This is a text question-
- Relevant Equations
- separation of variables
first order differential equation- integrating factor...
My thinking is two-fold, firstly, i noted that we can use separation of variables; i.e
##\dfrac{dy}{y}= \sec^2 x dx##
on integrating both sides we have;
##\ln y = \tan x + k##
##y=e^{\tan x+k} ##
now i got stuck here as we cannot apply the initial condition ##y(\dfrac {π}{4})=-1##
Secondly on using;
##\dfrac{dy}{dx}+ P(x)y=q(x)##
i have
##\dfrac{dy}{dx}-\sec^2 x=0##
i.f= ##e^{-\int sec^2x dx} =e^{-\tan x}##
therefore,
##(e^{-\tanx }⋅y)' =0## on integration, we shall have;
##(e^{-\tan x} ⋅y) =k## now using the initial condition, ##y(\dfrac {π}{4})=-1##
we have, ##k=-\dfrac{1}{e}##
thus,
##y=e^{\tan x} ⋅k=\left[ e^{\tan x} ⋅-\dfrac{1}{e}\right]=-e^{\tan x-1}##
i do not have solution to the problem...your insight is welcome...
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