Find the general solution of the given differential equation. cosx (dy/dx) + (sinx)y = 1

Solution:

A differential equation is an equation that contains at least one derivative of an unknown function, either an ordinary derivative or a partial derivative. 

Given, cos x (dy/dx) + (sin x)y = 1

Dividing by cos x,

(dy/dx) + y (sin x / cos x) = (1/cos x)

(dy/dx) + y tan x = sec x

The above linear differential equation of the type:

(dy/dx) + Py = Q

Where, P = tan x

Q = sec x

On integrating,

∫P dx = ∫tanx dx = log sec x

e^{∫P dx} = e^{log sec x} = sec x

We know, ye^{∫P dx} = ∫Q e^{∫P dx} dx + c

Y sec x = ∫sec x × sec x × dx + c

Y sec x = ∫sec²x dx + c

Y sec x = tan x + c

Dividing by sec x

y = sin x + c cos x

Therefore, the general solution is y = sin x + c cos x.

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Find the general solution of the given differential equation. cosx (dy/dx) + (sinx)y = 1

Summary:

The general solution of the given differential equation cos x (dy/dx) + (sin x)y = 1 is y = sin x + c cos x.