Separable DEs with Initial Conditions

This problem focuses on solving a first-order separable differential equation with an initial condition, a staple topic in early calculus and differential equations courses. The setup involves recognizing that the given equation can be written in a form where all terms involving the dependent variable are on one side and all terms involving the independent variable are on the other. The candidate must then integrate both sides and use the initial condition to identify the particular solution from the family of general solutions.

The question leans on fluency with separation of variables, basic integration, and handling exponential-type solutions that arise from linear differential equations. It also tests whether the candidate remembers to include the constant of integration and then correctly determines it from the initial value. Interviewers watch for clean algebraic manipulation, correct use of differential notation, and an understanding that the initial condition is essential to pin down a unique solution rather than leaving the answer as an arbitrary constant family.

What it tests

Separable differential equations rely on the ability to rewrite the derivative as a product (or quotient) of a function of `x` and a function of `y`, allowing each variable to be isolated on opposite sides of the equation. This structure enables direct integration with respect to each variable, leveraging the fundamental theorem of calculus. The reason this works is that integration is the inverse of differentiation, so if the rate of change of `y` with respect to `x` can be decomposed into independent influences of `x` and `y`, their cumulative effects can be 'summed up' separately. This pattern is not unique to any particular function: whenever you can write $dy/dx = f(x)g(y)$, you can always separate and integrate. The key is recognizing when such a separation is possible, which sometimes requires algebraic manipulation or substitution.