Unit 12 · Lesson 03 · Week 33 · Differential Equations & Growth Models
Separation of Variables
A differential equation specifies a relationship between a quantity and its rate. Slope fields visualize local directions; initial conditions select one solution from a family. This lesson emphasizes separation of variables through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in separation of variables.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do differential equations encode change, and how do slope fields and initial conditions determine model behavior?
Visual model
A differential equation specifies a relationship between a quantity and its rate. Slope fields visualize local directions; initial conditions select one solution from a family.
\frac{dy}{dx}=F(x,y)y_{n+1}=y_n+F(x_n,y_n)\Delta x
Worked reasoning
Misconception check: A slope field is not the graph of dy/dx as a separate function; each segment belongs at an (x,y) location.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Separation of Variables
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
For dy/dx=x−y, find the slope at (2,1).
Answer: 1
Reasoning: Substitute x=2,y=1.
Reasoning: Substitute x=2,y=1.
Verify y=Ce^x solves y'=y.
Answer: Yes
Reasoning: The derivative of Ce^x equals Ce^x.
Reasoning: The derivative of Ce^x equals Ce^x.
Solve dy/dx=2y by separation.
Answer: y=Ce^(2x)
Reasoning: dy/y=2dx, then integrate.
Reasoning: dy/y=2dx, then integrate.
Use y(0)=3 in the previous family.
Answer: y=3e^(2x)
Reasoning: The initial condition gives C=3.
Reasoning: The initial condition gives C=3.
One Euler step for y'=x+y from (0,1) with Δx=0.1.
Answer: y₁=1.1
Reasoning: Slope is 1, so y₁=1+1(0.1).
Reasoning: Slope is 1, so y₁=1+1(0.1).
What does an equilibrium solution require?
Answer: The derivative equals zero.
Reasoning: The quantity remains constant.
Reasoning: The quantity remains constant.