# \n\\( \\frac { d y } { d \\theta } = 4 y ^ { 2 } \\sec ^ { 2 } ( 2 \\theta ) \\quad y \\left( \\frac {…

Explanation:

Step1: Separate variables

Separate the variables in the differential equation (\frac{dy}{d\theta}=4y^{2}\sec^{2}(2\theta)). We get (\frac{dy}{y^{2}} = 4\sec^{2}(2\theta)d\theta).

Step2: Integrate both sides

Integrate (\int y^{- 2}dy=\int4\sec^{2}(2\theta)d\theta). For the left - hand side, using the power rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we have (\int y^{-2}dy=-\frac{1}{y}+C_1). For the right - hand side, let (u = 2\theta), then (du=2d\theta) and (\int4\sec^{2}(2\theta)d\theta=4\times\frac{1}{2}\int\sec^{2}(u)du). Since (\int\sec^{2}(x)dx=\tan(x)+C), we get (2\tan(2\theta)+C_2). So, (-\frac{1}{y}=2\tan(2\theta)+C).

Step3: Use the initial condition (y(\frac{\pi}{8}) = 1)

Substitute (\theta=\frac{\pi}{8}) and (y = 1) into (-\frac{1}{y}=2\tan(2\theta)+C). (-1=2\tan(\frac{\pi}{4})+C). Since (\tan(\frac{\pi}{4}) = 1), we have (-1=2\times1+C), then (C=-3). The particular solution is (-\frac{1}{y}=2\tan(2\theta)-3), or (y=\frac{1}{3 - 2\tan(2\theta)}).

Step4: Evaluate (y(\frac{3\pi}{8}))

Substitute (\theta=\frac{3\pi}{8}) into (y=\frac{1}{3 - 2\tan(2\theta)}). (2\theta=\frac{3\pi}{4}), and (\tan(\frac{3\pi}{4})=-1). (y(\frac{3\pi}{8})=\frac{1}{3-2\times(-1)}=\frac{1}{5}).

Answer:

(\frac{1}{5})