Solve the following differential equation: \cos^2 x \frac{dy}{dx} + y = tan x

CBSE Class 12 Math 2008 Solved Paper - Question 19

Question : 19 of 29

Marks: +1, -0

Solve the following differential equation:

cos^2 x {dy}/{dx}

+ y = tan x

Solution:

cos^2 x {dy}/{dx}

+ y = tan x

{dy}/{dx} + sec^2 x.y

= tan x .

sec^2

x
It is a linear differential equation of the first order.
Comparing it with

{dy}/{dx}

+ Py = Q, we get
P =

sec^2

x and Q = tan x.

sec^2

x
Integration factor =

e^{∫Pdx}

e^{∫sec^2xdx}

e^{tanx}

The solution of the given linear differential equation is given as:
y

e^{tanx}

= ∫ tan x .

sec^2

x .

e^{tanx}

dx + C
Let tan x = t ⇒

sec^2

x dx = dt

ye^t

= ∫ t .

e^t

. dt + C

ye^t

t.e^t

- ∫ 1.

e^t

dt + C

ye^t

t.e^t - e^t

+ C

ye^t

e^t

(t - 1) + C

ye^{tanx}

e^{tanx}

(tan x - 1) + C
y = tan x - 1 +

Ce^{-tanx}

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