Word Count : 559

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We are asked to solve the Riccati equation manually: $$ \frac{dy}{dx} = \frac{2\cos^2 x - \sin^2 x + y^2}{2 \cos x}, \quad y_1(x) = \sin x $$ --- ### Step 1: Recall the substitution for Riccati equations A Riccati equation has the general form: $$ \frac{dy}{dx} = q_0(x) + q_1(x) y + q_2(x) y^2 $$ If a particular solution $y_1(x)$ is known, we can reduce it to a linear equation via the substitution: $$ y = y_1 + \frac{1}{v} \quad \Rightarrow \quad \frac{dy}{dx} = \frac{dv/dx}{-v^2} + \frac{dy_1}{dx} $$ --- ### Step 2: Apply _________ ______ _______ ________ _________ ____ ________.
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