### Related Formula Standard first-order linear differential equation structure: fracdydx + P(x)y = Q(x)$\frac{dy}{dx} + P(x)y = Q(x)$ textIntegrating Factor (I.F.) = e^int P(x)dx$\text{Integrating Factor (I.F.)} = e^{\int P(x)dx}$ ### Step 1: Reduce into Standard Format Divide the full expression by 2cos x$2\cos x$ [cite: 3411, 4085]: fracdydx = frac2sin x cos x2cos x - frac4ysin x2cos x$\frac{dy}{dx} = \frac{2\sin x \cos x}{2\cos x} - \frac{4y\sin x}{2\cos x}$ fracdydx + 2ytan x = sin x$\frac{dy}{dx} + 2y\tan x = \sin x$ ### Step 2: Integrating Factor & Solution Compute the integrating multiplier : textI.F. = e^int 2tan x dx = e^2ln|sec x| = sec^2 x$\text{I.F.} = e^{\int 2\tan x dx} = e^{2\ln|\sec x|} = \sec^2 x$ Write general integration solution path : y cdot sec^2 x = int sin x cdot sec^2 x dx = int sec x tan x dx = sec x + C$y \cdot \sec^2 x = \int \sin x \cdot \sec^2 x dx = \int \sec x \tan x dx = \sec x + C$ y = cos x + Ccos^2 x$y = \cos x + C\cos^2 x$ ### Step 3: Boundary Evaluation Apply the initialization condition yleft(fracpi3right) = 0$y\left(\frac{\pi}{3}\right) = 0$ [cite: 3412, 4087]: 0 = cosleft(fracpi3right) + Ccos^2left(fracpi3right) Rightarrow 0 = frac12 + Cleft(frac14right) Rightarrow C = -2$0 = \cos\left(\frac{\pi}{3}\right) + C\cos^2\left(\frac{\pi}{3}\right) \Rightarrow 0 = \frac{1}{2} + C\left(\frac{1}{4}\right) \Rightarrow C = -2$ Thus, the solution is y = cos x - 2cos^2 x$y = \cos x - 2\cos^2 x$ . Find derivative yprime$y\prime$ : yprime = -sin x + 4cos x sin x = -sin x + 2sin 2x$y\prime = -\sin x + 4\cos x \sin x = -\sin x + 2\sin 2x$ ### Step 4: Target Calculation Evaluate components at x = fracpi4$x = \frac{\pi}{4}$ : yleft(fracpi4right) = frac1sqrt2 - 2left(frac12right) = frac1sqrt2 - 1$y\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} - 2\left(\frac{1}{2}\right) = \frac{1}{\sqrt{2}} - 1$ yprimeleft(fracpi4right) = -frac1sqrt2 + 2sinleft(fracpi2right) = -frac1sqrt2 + 2$y\prime\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} + 2\sin\left(\frac{\pi}{2}\right) = -\frac{1}{\sqrt{2}} + 2$ yprimeleft(fracpi4right) + yleft(fracpi4right) = left(-frac1sqrt2 + 2right) + left(frac1sqrt2 - 1right) = 1$y\prime\left(\frac{\pi}{4}\right) + y\left(\frac{\pi}{4}\right) = \left(-\frac{1}{\sqrt{2}} + 2\right) + \left(\frac{1}{\sqrt{2}} - 1\right) = 1$ ### Pattern Recognition Linear standard layout conversions depend entirely on clear integrating factor reductions. Remember int tan x dx = ln|sec x|$\int \tan x dx = \ln|\sec x|$ clearly to safely output exact matching polynomial definitions. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Differential Equations