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Question
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Solution
3ex tanydx + (1 +ex) sec2 ydy = 0
Divide by tany(1+ex)
`(3e^x)/(1+e^x)dx+(sec^2y)/(tany)dy=0`
`int(3e^x)/(1+e^x)dx+int(sec^2y)/(tany)dy=0`
Put tan y = t
dt = sec2y
`3 log|1+e^x|+int1/tdt=c`
`3log|1+e^x|+log|t|=c`
`3log|1+e^x|+log|tany|=c ................(1)`
put x=0 and y=pi in (1)
`3log|1+e^0|+log|0|=c`
`3log|2|=c................(2)`
`3log|1+e^x|+log|tany|=3log|2| `
`3log|1+e^x|+log|tany|-3log|2|=0`
`3(log|1+e^x|-log|2|)+log|tany|=0`
`3log((1+e^x)/2)+log|tany|=0`
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