Florida B.E.S.T. MA.912.C.5.2

MathGrades 9–12Apply integrals to solve problems.

The Standard

Solve separable differential equations.

Florida B.E.S.T. Standards for Mathematics

Teacher's field guide

What Students Need to Do: Students rewrite a differential equation so all y terms are with dy and all x terms are with dx. They integrate both sides and solve for the function when possible. They use initial values to find constants and check the solution.
What Mastery Looks Like: Students identify when an equation is separable and place x and y terms on opposite sides. They integrate, apply an initial condition, and state any equilibrium solutions. They verify the result by differentiation.
Common Misconceptions: Students may separate terms incorrectly or treat dy/dx as an ordinary fraction without preserving the equation. They often forget the constant of integration, mishandle absolute values, or lose equilibrium solutions when dividing by a variable expression.
How to Assess It: Give the prompt: “Solve dy/dx = xy with y(0) = 3, and verify your solution by differentiation.” Look for y = 3e^(x²/2) and a correct check.

Lesson moves

Give groups cards showing differential equations, separated forms, antiderivatives, and solutions, then have them match each complete solution chain.
Ask students to explain why dividing by y in dy/dx = y(1-y) can cause a solution to disappear.
Run a four-station relay where teams separate, integrate, apply an initial condition, and verify one equation at successive stations.
Model coffee cooling with dT/dt = -0.1(T-22) and T(0) = 90, then predict the temperature after 10 minutes.

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