You encounter the equation dy/dx = x + y. Can you solve it by separation of variables?
AYes — move all x terms to the right and all y terms to the left
BNo — the right side is a sum (x + y), not a product f(x)·g(y), so the variables cannot be separated
CYes — integrate both sides directly with respect to x
DNo — only first-order equations with constant coefficients can be separated
Separation of variables requires the equation to have the form dy/dx = f(x)·g(y) — the right side must factor into a product of a pure function of x and a pure function of y. The sum x + y cannot be factored this way: there is no way to write it as f(x)·g(y). This is the structural test for separability. An equation like dy/dx = xy (a product) is separable; dy/dx = x + y (a sum) is not.
Question 2 Multiple Choice
After separating and integrating the equation dy/dx = 2xy, a student obtains ln|y| = x² + C. What does the constant C represent geometrically?
AThe domain over which the solution is defined
BThe parameter that selects one particular solution from the family of curves, determined by an initial condition
CThe type of integration technique used on the left side
DWhether the solution is expressed implicitly or explicitly
The general solution ln|y| = x² + C (or equivalently y = Ae^(x²)) is a family of curves filling the xy-plane, one for each value of C. An initial condition y(x₀) = y₀ picks the specific curve passing through the point (x₀, y₀), determining C. Without an initial condition, C remains a free parameter and the solution is the entire family. C is not an artifact of the integration method — it is the essential parameter that distinguishes one particular solution from all the others.
Question 3 True / False
A separable differential equation can yield a solution expressed as an implicit relation between x and y rather than as an explicit formula y = h(x).
TTrue
FFalse
Answer: True
After integrating both sides of a separated equation, you typically have an implicit equation like F(y) = G(x) + C. Sometimes this can be solved explicitly for y; sometimes it cannot. Both implicit and explicit forms are valid complete solutions. For example, separating and integrating dy/dx = y/x gives ln|y| = ln|x| + C, which simplifies to y = Ax. But more complex integrals may leave an implicit equation that cannot be inverted.
Question 4 True / False
The general solution to a separable ODE is a unique curve determined by the equation alone.
TTrue
FFalse
Answer: False
The general solution is a family of curves, parameterized by the constant of integration C. The equation alone does not pick one — it describes the entire collection of solutions consistent with the differential relationship. Only an initial condition (a specified point the solution must pass through) pins down C and selects one particular solution from the family.
Question 5 Short Answer
What does it mean for a differential equation to be 'separable,' and why does that structural property allow integration to solve it?
Think about your answer, then reveal below.
Model answer: A separable equation has the form dy/dx = f(x)·g(y): the right side factors into a pure function of x times a pure function of y. This means you can divide both sides by g(y) and multiply by dx to get (1/g(y)) dy = f(x) dx — all y-expressions on the left, all x-expressions on the right. Each side can now be integrated independently, converting the differential equation into two standard integration problems. The factored structure is the key: if the right side cannot be written as a product of separate functions, the variables cannot be separated and this technique does not apply.
Separability is a structural property of the equation, not a technique. The technique (separate and integrate) only works because the structure exists. Checking separability is always the first step, and recognizing when an equation is NOT separable (e.g., dy/dx = x + y) is as important as solving those that are.