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Find the general solution for \(\frac {dy}{dx}\) + 2y tan x = sin x differential equations.

dy\dx + 2y tan x = sin x

1 Answer

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Best answer

Given Differential Equation :

\(\frac{dy}{dx}\) + 2y tan x = sin x

Formula :

i) \(\int\) tan x dx = log |sec x|

ii) alog b = log ba

iii) aloga b = b

iv) \(\int\) \((\frac{-1}{x^2})\) dx = \(\frac{1}{x}\)

v) General solution : 

For the differential equation in the form of

\(\frac{dy}{dx} \, + Py \, =Q\)

General solution is given by,

y. (I.F.) = \(\int\) Q.(I.F.) dx + c

Where, integrating factor,

I.F. = \(e^{\int p\, dx}\)

Given differential equation is

\(\frac{dy}{dx}\) + 2y tan x = sin x ....eq(1)

Equation (1) is of the form

\(\frac{dy}{dx} \, + Py \, =Q\)

where, P = 2tan x and Q = sin x

Therefore, integrating factor is

General solution is

Substituting I in eq(2),

Multiplying above equation by cos2x,

Therefore, general solution is

y = cos x + c(cos2 x)

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