AP Calculus AB Differential Equations and Slope Fields Lesson (Free PDF)
This differential equations lesson for AP Calculus AB covers differential equations and slope fields (Common Core AP CALC AB 7.1, AP CALC AB 7.2, AP CALC AB 7.3, AP CALC AB 7.4, AP CALC AB 7.5, AP CALC AB 7.6, AP CALC AB 7.7), with a free PDF to print.
A differential equation relates a function to its derivative. A slope field shows dy/dx at many points, and Euler’s method approximates a solution with tangent line steps. Separable equations are solved by separating variables, integrating, and using an initial condition. If dy/dt = ky, then y = y₀e^(kt).
AP Calculus AB · Common Core State Standards: AP CALC AB 7.1, AP CALC AB 7.2, AP CALC AB 7.3, AP CALC AB 7.4, AP CALC AB 7.5, AP CALC AB 7.6, AP CALC AB 7.7
Key facts in this differential equations lesson
- Verify a solution by substituting y and y′ into the equation.
- A slope field shows dy/dx at grid points; solution curves follow the segments.
- Euler: y_new = y_old + h · (dy/dx at the old point), x_new = x_old + h.
- dy/dt = ky has the general solution y = Ce^(kt); growth if k is positive, decay if k is negative.
What is a differential equation?
A differential equation relates a function to its derivative, such as \(\frac{dy}{dx}=x-y\). A solution is a function that makes the equation true. You can verify a proposed solution by substituting it and its derivative into the equation. For example, \(y=e^{3x}\) solves \(\frac{dy}{dx}=3y\) because \(\frac{d}{dx}e^{3x}=3e^{3x}\).
What is a slope field?
A slope field draws, at many points \((x,y)\), a short segment with slope \(\frac{dy}{dx}\) evaluated there. Solution curves follow the segments like a river following arrows.
- Where \(\frac{dy}{dx}=0\) the segments are horizontal.
- If the formula depends only on \(x\), segments in a vertical line are parallel.
- If it depends only on \(y\), segments in a horizontal line are parallel.
What is Euler’s method?
Euler’s method approximates a solution by following tangent lines. Starting at \((x_{0},y_{0})\) with step \(h\): \(y_{n+1}=y_{n}+h\cdot\frac{dy}{dx}\Big|_{(x_{n},y_{n})}\) and \(x_{n+1}=x_{n}+h\). For \(\frac{dy}{dx}=x+y\) with \(y(0)=1\) and \(h=0.5\), \(y_{1}=1+0.5(0+1)=1.5\).
How do you solve by separation of variables?
Example. Solve \(\frac{dy}{dx}=\frac{x}{y}\) with \(y(0)=2\). Separate: \(y\,dy=x\,dx\). Integrate: \(\frac{y^{2}}{2}=\frac{x^{2}}{2}+C\). Use \(y(0)=2\) to get \(C=2\), so \(y^{2}=x^{2}+4\) and \(y=\sqrt{x^{2}+4}\) (positive because \(y(0) > 0\)).
To check a separated solution, differentiate it. For \(y=\sqrt{x^{2}+4}\), \(\frac{dy}{dx}=\frac{x}{\sqrt{x^{2}+4}}=\frac{x}{y}\), which is the original equation, and \(y(0)=\sqrt{4}=2\) matches the initial condition. Always report the interval where the solution is valid: for \(\frac{dy}{dx}=y^{2}\) with \(y(0)=1\) the solution \(y=\frac{1}{1-x}\) exists only for \(x < 1\).
What are exponential growth and decay models?
If a quantity changes at a rate proportional to itself, \(\frac{dy}{dt}=ky\), then \(y=y_{0}e^{kt}\). With \(k > 0\) this is growth; with \(k < 0\), decay. For a population \(\frac{dP}{dt}=0.03P\) with \(P(0)=500\), \(P=500e^{0.03t}\), so \(P(10)=500e^{0.3}\approx 675\).
Two features help you recognize exponential models. First, the growth rate \(\frac{dP}{dt}\) is proportional to \(P\), so a larger population grows faster. Second, the time to double does not depend on the starting size: the doubling time is always \(\frac{\ln 2}{k}\). For decay with half-life \(T\), the constant is \(k=\frac{\ln 2}{T}\).
Differential equations lesson: questions and answers
What is a differential equation?
It is an equation that involves a function and its derivatives. A solution is a function that satisfies the equation, often with an initial condition that picks out one particular solution.
How do you read a slope field?
Each short segment shows the slope of a solution curve at that point. Follow the segments to sketch a solution through a given starting point.
How does Euler’s method work?
Start at the initial point, compute the slope from the differential equation, move horizontally by the step h along the tangent line, and repeat from the new point.
What is separation of variables?
It rewrites dy/dx = f(x)g(y) as dy/g(y) = f(x) dx so that each side can be integrated separately. Remember to add the constant C and use the initial condition.
What is the solution of dy/dt = ky?
The solution is y = Ce^(kt), where C = y(0). A positive k gives exponential growth and a negative k gives exponential decay.
How to use this differential equations lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the differential equations lesson as a PDF with the button below and keep it with your notes.
Download the differential equations lesson as a PDF with the button below and keep it with your notes.
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