Separation of variables
Also known as: separable differential equations
Separation of variables is a method for solving differential equations by moving all terms with one variable to one side and all terms with the other variable to the opposite side, then integrating both sides. It works when dy/dx can be written as a function of x times a function of y.
A first-order differential equation is separable when it can be rearranged into the form dy/dx = g(x)h(y) — the x pieces and the y pieces factor apart. Once separated, the equation becomes (1/h(y)) dy = g(x) dx, and integrating each side with respect to its own variable produces a relationship between x and y with no derivative in it.
Work through dy/dx = xy. Divide by y and multiply by dx to get (1/y) dy = x dx. Integrating both sides gives ln|y| = x²/2 + C. Exponentiating turns this into y = Ce^(x²/2), where the arbitrary constant absorbs the sign and the exponential of the original constant. Only one constant of integration is needed, conventionally written on the x side.
The result so far is the general solution, a family of curves. Adding an initial condition such as y(0) = 3 pins down C and yields the particular solution — here C = 3, so y = 3e^(x²/2). Substitute the initial condition as soon as you have the general solution rather than after further algebra, since it usually simplifies the work.
The most important application is exponential growth and decay. The equation dy/dt = ky says a quantity changes at a rate proportional to its current size, and separating variables gives y = y₀e^(kt) — the model behind population growth, radioactive decay, and continuously compounded interest. Newton's law of cooling, dT/dt = k(T − T_s), separates the same way, except there it is the temperature *difference* from the surroundings that decays exponentially. A slope field offers a visual check: sketching the general solution over the field should show the curves following the plotted slopes.
AP Calculus AB devotes a section of its differential equations unit to separation of variables, and free-response questions regularly ask you to separate, integrate, apply an initial condition, and interpret the resulting exponential model in context. Points are commonly lost by forgetting the constant of integration or by failing to solve explicitly for y when the question asks for y as a function of x.
Key takeaways
- A differential equation is separable when dy/dx factors into a function of x times a function of y.
- Move each variable to its own side, then integrate both sides independently.
- One constant of integration suffices, and an initial condition converts the general solution to a particular solution.
- dy/dt = ky separates to give the exponential model y = y₀e^(kt).
- AP Calculus AB free-response questions expect an explicit solution for y plus contextual interpretation.