Solve the differential equation: `cos^2x(dy)/(dx)+y=tanx`

Explanation

Given

`cos ^{2} x \cdot \frac{d y}{d x}+y=\tan x `

`frac{d y}{d x}+y \sec ^{2} x=\tan x \cdot \sec ^{2} x`

Given differential equation is a linear differential equation `frac{d y}{d x}+p y=Q`

`text { I.F. }=e^{\int P d x}=e^{\int \sec ^{2} x d x}=e^{\tan x}`

`\therefore \quad` Solution is given by

`e^{\tan x} y=\int \tan x \cdot \sec ^{2} x \cdot e^{\tan x} d x`

Let `I=\int \tan x \cdot \sec ^{2} x \cdot e^{\tan x} d x`

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