Differential Equation
An equation involving a function and its derivatives.
Example: dy/dx = 2x has solution y = x^2 + C.
Adaptive
Learn Differential Equations
Read the notes, then try the practice. It adapts as you go.When you're ready.
Session Length
~15 min
Adaptive Checks
14 questions
Transfer Probes
8
Lesson Notes
Differential equations relate a function to its derivatives. This topic covers separable DEs, slope fields, Euler method, and exponential models for AP Calculus AB Unit 7.
Key concepts in this area include Differential Equation, Separable DE, Slope Field, and Euler Method. Differential Equation refers to an equation involving a function and its derivatives. Separable DE, meanwhile, involves a DE writable as g(y)dy = f(x)dx.
By studying differential equations, learners develop the ability to verify solutions by substitution and solve separable DEs analytically. These skills build analytical thinking and prepare students for more advanced work in Mathematics.
You'll be able to:
- Verify solutions by substitution
- Solve separable DEs analytically
- Sketch and interpret slope fields
- Apply Euler method
- Model exponential growth/decay
One step at a time.
Interactive Exploration
Adjust the controls and watch the concepts respond in real time.
Key Concepts
Separable DE
A DE writable as g(y)dy = f(x)dx.
Example: dy/dx = xy separates to (1/y)dy = x dx.
Slope Field
Graphical representation of dy/dx at each point.
Example: For dy/dx = x-y, plot segments at grid points.
Euler Method
Numerical: y_{n+1} = y_n + h*f(x_n, y_n).
Example: h=0.1 from (0,1) using dy/dx=y gives y(0.1)=1.1.
Exponential Growth/Decay
dy/dt = ky => y = y_0*e^{kt}.
Example: dy/dt=0.05y, y(0)=100 => y=100e^{0.05t}.
Initial Value Problem
DE + initial condition y(x_0)=y_0.
Example: dy/dx=2x, y(0)=3 => y=x^2+3.
General vs Particular Solution
General has C. Particular fixes C via IC.
Example: y=x^2+C vs y=x^2+3.
Explore your way
Choose a different way to engage with this topic — no grading, just richer thinking.
Explore your way — choose one:
Concept Map
See how the key ideas connect. Nodes color in as you practice.
Worked Example
Walk through a solved problem step-by-step. Try predicting each step before revealing it.
Adaptive Practice
This is guided practice, not just a quiz. Hints and pacing adjust in real time.
Small steps add up.
What you get while practicing:
- Math Lens cues for what to look for and what to ignore.
- Progressive hints (direction, rule, then apply).
- Targeted feedback when a common misconception appears.
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