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

CBSE Class 12 Math 2011 Solved Paper - Question 19

Question : 19 of 29

Marks: +1, -0

Solve the following differential equation:

cos^2x {dy}/{dx}

+ y = tan x

Solution:

cos^2x {dy}/{dx}

+ y = tan x
⇒

{dy}/{dx}

sec^2

x.y =

sec^2

x tan x
This equation is in the form of

{dy}/{dx}

+ py = Q
here p =

sec^2

x and Q =

sec^2

x tan x
Integrating Factor , I.F =

e^{∫pdx}

e^{∫sec^2 xdx}

e^{tanx}

The general solution can be given by
y (I.F.) = ∫ (Q × I.F.) dx + C ... (1)
Let tanx = t
⇒

d/{dx}

(tan x) =

{dt}/{dx}

⇒

sec^2

x =

{dt}/{dx}

⇒

sec^2

x dx = dt
Therefore, equation 1 becomes :

y.e^{tanx}

= ∫

(e^t . t)

dt
⇒ y .

e^{tanx}

= ∫

(e^t . t)

dt + C
⇒ y .

e^{tanx}

= t . ∫

e^t

dt - ∫

(d/{dt}(t) . ∫e^t)dt

+ C
⇒ y .

e^{tanx}

t.e^t-∫e^t dt

+ C
⇒ y .

e^{tanx}

t.e^t -e^t

+ C
⇒ y .

e^{tanx}

= (t - 1)

e^t

+ C
⇒ y .

e^{tanx}

= (tan x - 1)

e^{tanx}

+ C
⇒ y = (tan x - 1) +

Ce^{-tanx}

, where C is an arbitary constant

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