Separation of variables and exponential models

Learning objectives

Solve a separable differential equation by separating the variables, integrating both sides and including a single constant of integration, to obtain the general solution

Find a particular solution that satisfies an initial condition, solving for the constant and then for y explicitly, and handle absolute values and logarithms correctly

State the domain of a particular solution, restricting it to the interval containing the initial condition on which the solution is defined and differentiable

Derive the solution y = y₀eᵏᵗ of dy/dt = ky and use it to model exponential growth and decay, including finding the constant k from given data

Set up and solve applied differential equations, such as Newton's law of cooling, population growth or a draining tank, and interpret the solution in context