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The question is to determine the solution to the following 1st order linear DE, along with the largest interval the solution is valid on:
cosx \frac{dy}{dx} + (sinx)y=1
Rewriting it shows it to be linear:
\frac{dy}{dx} + (tanx)y = secx
The intergrating factor is: e^{\int{tanx dx}} = e^{-ln|cosx|} = secx
Multiplying both sides of the DE by the integrating factor, and rewriting the LHS as a derivative of the product of the integrating factor and y:
\frac{d}{dx}[(secx)y]= sex^{2}x
(secx)y = tanx+c
y = sinx + c(cosx)
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Now how do I determine the interval?
cosx \frac{dy}{dx} + (sinx)y=1
Rewriting it shows it to be linear:
\frac{dy}{dx} + (tanx)y = secx
The intergrating factor is: e^{\int{tanx dx}} = e^{-ln|cosx|} = secx
Multiplying both sides of the DE by the integrating factor, and rewriting the LHS as a derivative of the product of the integrating factor and y:
\frac{d}{dx}[(secx)y]= sex^{2}x
(secx)y = tanx+c
y = sinx + c(cosx)
------------------
Now how do I determine the interval?