Differential Equations

To solve the given differential equation:$(\sin y \cos^2 y - x \sec^2 y) dy = (\tan y) dx,$we start by rewriting it in the standard linear form:$\frac{dx}{dy} + \frac{x}{\sin y \cos y} = \cos^3 y .$Recognizing this as a linear differential equation in $x$, we determine the integrating facCompute the Integrating Factor (IF):$IF = e^{\int \frac{1}{\sin y \cos y} dy} .$Solve the Integral:$\int \frac{1}{\sin y \cos y} dy = 2 \int \csc 2y dy .$Solving this integral gives:$e^{\int \csc 2y dy} = e^{\log |\csc 2y - \cot 2y|} = \csc 2y - \cot 2y .$Thus, the integrating factor simplifies to:$IF = \frac{1 - \cos 2y}{\sin 2y} = \frac{2 \sin^2 y}{2 \sin y \cos y} = \tan y .$Apply the Integrating Factor:Multiply the entire linear differential equation by this integrating factor:$x \times \tan y = \int \cos^3 y \times \tan y dy$Integrate the Right-Hand Side:Substitute $\cos^2 y = 1 - \sin^2 y$. Then:$\int \cos^3 y \tan y dy = \int \cos^2 y \sin y dy .$Perform the integration:$= -\frac{\cos^3 y}{3} + C .$Resulting General Solution:Therefore, the general solution to the differential equation is:$3 x \tan y + \cos^3 y = C .$