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 If   \frac{\mathrm{d} y}{\mathrm{d} x}+\frac{3}{\cos ^2x }y=\frac{1}{\cos ^2x },x\epsilon \left ( \frac{-\pi }{3},\frac{\pi }{3} \right ), ,and y\left ( \frac{\pi }{4} \right )= \frac{4}{3},

then y(-\pi/4) equals 

  • Option 1)

     

    1/3+e^6

  • Option 2)

     

    1/3

  • Option 3)

     

    -4/3

  • Option 4)

     

    1/3+e^3

Answers (1)

best_answer

 

Linear Differential Equation -

\frac{dy}{dx}+Py= Q

- wherein

P, Q are functions of x alone.

 

 

Linear Differential Equation -

Multiply by e^{SPdx}  which is the Integrating factor

- wherein

P is the function of x alone

\frac{dy}{dx}+ 3sec^{2}x.y = sec^{2}x

From the concept

I.F. = e^{3\int sec^{2}xdx} = e^{3tanx}

or

y.e^{3tanx} = \frac{1}{3}e^{3tanx} + c ........... (1)

given, y\left ( \frac{\pi}{4} \right ) = \frac{4}{3}

\therefore \frac{4}{3} e^{3} = \frac{1}{3} e^{3} + c

c = e^{3}

Now put x = -\frac{\pi}{4} \ in \ (1)

y.e^{-3} =\frac{1}{3}e^{-3} + e^{3}

\therefore y =\frac{1}{3} + e^{6}

 


Option 1)

 

1/3+e^6

Option 2)

 

1/3

Option 3)

 

-4/3

Option 4)

 

1/3+e^3

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