If and equal

Engineering

Mathematics

Linear Differential Equation

Question

If $\frac{dy}{dx} + \frac{3}{\cos^{2} x} y = \frac{1}{\cos^{2} x}, x \in (\frac{- π}{3}, \frac{π}{3})$ and $y (\frac{π}{4}) = \frac{4}{3}, then y (- \frac{π}{4})$ equal

$\frac{1}{3}$

$\frac{1}{3} + e^{6}$

$\frac{1}{3} + e^{3}$

$- \frac{4}{3}$

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$\frac{dy}{dx} + (3 \sec^{2} x) y = \sec^{2} H$

This is linear differential equation

$Hence y . e^{\int 3 tanx} . \sec^{2} xdx$

$\Rightarrow {ye}^{3 tanx} = \frac{e^{3 tanx}}{3} + c$

$\Rightarrow y = {Ce}^{- 3 tanx} + \frac{1}{3}$

$Giveny y (\frac{π}{4}) = \frac{4}{3} \Rightarrow \frac{4}{3} = {Ce}^{- 3} + \frac{1}{3}$

⇒ C = e³

^{$Hence y (\frac{π}{4}) = e^{3} . e^{3} + \frac{1}{3} = e^{6} + \frac{1}{3}$}