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प्रश्न
Solve the differential equation `cos^2 x dy/dx` + y = tan x
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उत्तर
`cos^2 x dy/dx` + y = tan x
∴ `dy/dx + y/(cos^2x) = tanx/(cos^2x)`
∴ `dy/dx + sec^2x.y` = tan x . sec2 x
The given equation is of the form
`dy/dx + Py` = Q,
Where P = sec2 x and Q = tan x. sec2 x
∴ I.F. = `e^(int Pdx) = e^(intsec^2x dx)` = etan x
∴ Solution of the given equation is
y(I.F.) = `int Q.(I.F.)dx + c`
∴ yetan x = `int tan x.sec^2x.e^(tanx)dx+ c`
Put tan x = t
∴ sec2x dx = dt
∴ yetan x = `int te^t dt + c`
= `tint e^t dt - int[d/dt (t) inte^tdt]dt + c`
= `te^t - int e^tdt + c`
= tet – et + c
∴ yetan x = etanx (tanx – 1) + c
∴ y = tan x – 1 + c.e–tanx
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