Find the general solution of the given differential equation. Give the largest i

fortdefruitI

fortdefruitI

Answered question

2021-10-23

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution.
cosxdydx+(sinx)y=1

Answer & Explanation

rogreenhoxa8

rogreenhoxa8

Skilled2021-10-24Added 109 answers

cosx dy  dx +sinxy=1
By dividing both sides cosx
 dy  dx +sinxcosxy=1cosx
This is the form of the differential equation.  dy  dx +P(x)y=f(x)
With P(x)=sinxcosx  and  f(x)=1cosx
P(x) is continuous on (π2,π2)
Find the integrating factor I(x)
I(x)=eP(x) dx =esinxcosx dx =eln|cosx|=1cosx
Add the integrating factor to the differential equation.
( dy  dx +sinxcosxy)1cosx=(1cosx)1cosx
1cosx dy  dx +sinxcos2xy=1cos2x
Write the left side in the form d dx [I(x)y]
d dx [ycosx]=sec2x
d[ycosx]=sec2x dx 
combine the two 
ycosx=sec2x dx 
ycosx=tanx+C
Solve for y
 

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