Find the value of y(-pi/4)

Question

Answer 1

\(\frac{dy}{dx}+\frac{3}{cos^2x}y=\frac1{cos^2x}\) 

I. F. = e^∫ρdx = e^{∫\(\frac{3}{cos^2x}dx\)} = e^{3∫\(sec^2x\) dx}

= e^3tanx

Complete solution is

y x I.F = ∫(I. F) x Q dx

⇒ y e^{3tan x} = ∫sec²x e^3tanxdx

= ∫e^3t df (By taking t = tan x, \(\therefore\) df = sec²x dx)

⇒ y e^3tanx = \(\frac{e^{3t}}{3}+c\) (By putting t = tan x)

\(\because\) y(π/4) = 4/3

\(\therefore\) 4/3 x e^3tan π/4 = \(\frac{e^{3 tan\pi/4}}3+c\) 

⇒ c = 4/3 e³ - e³/3 = e³

\(\therefore\) y e^{3tan x} = \(\frac{e^{3 tan\,x}}3+e^3\) 

⇒ y = 1/3 + e³ e^{-3 tan x} = 1/3 + e^{3(1- tan x)}

Now, y (-π/4) = 1/3 + e^{3(1-tan(-π/4))} = 1/3 + e^{3(1 + 1)}

= 1/3 + e⁶

option (A) is correct.

Comments

Thanks for the answer
Best solution