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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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