Kerala KEAM 5 June 2024 Solved Paper

We are given the differential equation: $$\frac{e^y}{dx} = x + 2$$First, rewrite it in the form: $$e^y \frac{dy}{dx} = x + 2$$Now, separate the variables $y$ and $x$: $$e^y dy = (x + 2) dx$$Integrate both sides: $$\int e^y \, dy = \int (x + 2) \, dx$$The integral of $e^y$ with respect to $y$ is $e^y$, and the integral of $(x + 2)$ with respect to $x$ is: $$\int (x + 2) \, dx = \frac{(x + 2)^2}{2} + C$$Thus, the solution is: $$e^y = \frac{(x + 2)^2}{2} + C$$Solving for $y$, we get: $$y = \log(x + 2) + C$$Thus, the correct answer is option (B): $$y = (x + 2) \log(x + 2) - x + C$$Thus, the correct answer is option (B). Quick Tip: For solving first-order differential equations, use the separation of variables method, then integrate each side. Don’t forget to solve for the dependent variable at the end.