Kerala KEAM 5 June 2024 Solved Paper

We are given the differential equation: $$\frac{dy}{\cos y} = dx$$We can separate the variables $y$ and $x$ as follows: $$\frac{dy}{\cos y} = dx \quad \Rightarrow \quad \int \frac{dy}{\cos y} = \int dx$$The integral of $\frac{1}{\cos y}$ is $\sec y$, and the integral of $dx$ is $x$. Thus, we have: $$\sec y = x + C$$Now, to solve for $y$, take the logarithm of both sides: $$\log | \sec y + \tan y | = x + C$$Thus, the correct answer is option (E): $$\log | \sec y + \tan y | = x + C$$Thus, the correct answer is option (E). Quick Tip: For integrals involving trigonometric functions like $\sec y$, use the identity $\sec^2 y - \tan^2 y = 1$ to simplify the expression if necessary. Also, remember to apply logarithms when dealing with functions of the form $\sec y + \tan y$.