Differential equation Problem: x dy y dx

[latex] \[ \textbf{Q1: Solve the differential equation:} \quad x \, dy = y \, dx \] \[ \textbf{Step 1: Rearranging the equation} \] We start by separating the variables \( x \) and \( y \). We divide both sides by \( y \) and \( x \): \[ \frac{dy}{y} = \frac{dx}{x} \] \[ \textbf{Step 2: Integration of both sides} \] Now, we integrate both sides of the equation. On the left side, we integrate with respect to \( y \), and on the right side, we integrate with respect to \( x \): \[ \int \frac{1}{y} \, dy = \int \frac{1}{x} \, dx \] The integrals are straightforward: \[ \ln |y| = \ln |x| + C \] where \( C \) is the constant of integration. \[ \textbf{Step 3: Solve for \( y \)} \] Next, we solve for \( y \). To do this, we exponentiate both sides of the equation to eliminate the logarithms: \[ e^{\ln |y|} = e^{\ln |x| + C} \] Using the properties of exponents, we can simplify: \[ |y| = e^{C} \cdot |x| \] Let \( e^C = C_1 \) (where \( C_1 \) is a new constant), so the equation becomes: \[ |y| = C_1 x \] \[ \textbf{Step 4: Final solution} \] Since \( |y| \) can be both positive and negative, we drop the absolute value and write the final general solution as: \[ y = C_1 x \] where \( C_1 \) is the constant of integration. This is the general solution to the differential equation \( x \, dy = y \, dx \).