Differential equations describe relationships between a function and its rate of change. In calculus, they are used to model motion, population growth, cooling, mixing, and many other changing quantities. This cheat sheet helps students connect symbolic equations, graphical slope fields, and solution curves.
It is especially useful for AP Calculus and precalculus students beginning differential equation models.
The most important idea is that a differential equation such as gives the slope of a solution curve at each point. A slope field shows many small tangent segments that guide the shape of possible solutions. Some differential equations can be solved by separating variables and integrating both sides.
Common models include exponential growth or decay, , and logistic growth, .
Key Facts
- A first-order differential equation has the form and gives the slope of at each point .
- A slope field is made by drawing a short line segment with slope at many points in the coordinate plane.
- A solution curve to must be tangent to the slope field segment at every point it passes through.
- A separable differential equation can be written as , then solved by integrating .
- The general solution of exponential growth or decay is , where is the initial amount when .
- If in , the model shows growth, and if , the model shows decay.
- The logistic differential equation has carrying capacity and grows fastest when .
- An initial condition such as selects one particular solution from a family of solutions.
Vocabulary
- Differential Equation
- An equation involving a function and one or more of its derivatives, such as .
- Slope Field
- A graph of short line segments showing the slope given by at many points.
- Solution Curve
- A curve whose tangent slope at every point matches the differential equation.
- Separation of Variables
- A method for solving some differential equations by placing all terms with and all terms with .
- Initial Condition
- A given value such as that determines a specific solution from the general solution.
- Equilibrium Solution
- A constant solution where the derivative is zero, such as .
Common Mistakes to Avoid
- Treating a slope field segment as a point value is wrong because the segment shows the local slope, not the height of the solution.
- Forgetting the constant of integration is wrong because solving gives a family of solutions until an initial condition is applied.
- Separating variables incorrectly is wrong because every factor containing must stay with and every factor containing must stay with .
- Drawing solution curves that cross is wrong for most basic first-order differential equations because one point cannot have two different solution directions.
- Assuming exponential growth for every population model is wrong because logistic models include a limiting carrying capacity .
Practice Questions
- 1 For , find the slope of the solution curve at the point .
- 2 Solve the separable differential equation with initial condition .
- 3 A population satisfies . Find the carrying capacity and the population size where growth is fastest.
- 4 Explain how a slope field can show whether solutions are increasing, decreasing, or approaching an equilibrium solution without solving the differential equation.
Understanding Differential Equations and Slope Fields
A slope field is more than a picture of one answer. It shows the local behavior of every possible answer at once. Start at an initial point, then follow the nearby segments in small steps.
This is the same basic idea used in numerical methods when an exact formula is difficult or impossible to find. Curves that begin at different heights usually remain distinct. For many well behaved equations, two solution curves cannot cross because a crossing point would require one point to have two different directions.
This helps students check sketches. A curve that cuts across the tiny segments at a sharp angle cannot be a valid solution.
Some horizontal levels have a special role. If every segment along a horizontal line is flat, a solution beginning there stays there. Such a constant solution is called an equilibrium.
Points near an equilibrium can move toward it, away from it, or behave differently on opposite sides. This tells you about stability without solving the equation. For example, a population level may return toward a long term limit after a small disturbance.
A balance point can be unstable if a small change causes the quantity to move farther away. Reading these patterns from a field is an important skill in calculus because real models often need qualitative predictions before exact calculations.
Separation of variables works only when all terms involving the dependent quantity can be placed on one side and all terms involving the independent quantity can be placed on the other. The integration step requires care. Each side gains a constant, but the two constants can be combined into one constant afterward.
Students often lose a constant solution by dividing by an expression that might equal zero. Check those cases before dividing. After finding a formula, use the initial condition to determine the remaining constant.
Then check the domain. A formula may fail at a value where a denominator becomes zero, where a logarithm is undefined, or where the original model stops making physical sense.
Growth models are useful only when their assumptions fit the situation. Exponential change assumes that the rate is proportional to the current amount. This can describe radioactive material, compound interest over a limited period, or early bacterial growth.
It does not include limits on food, space, money, or resources. Logistic growth includes a limiting population level, so its graph often has an S shape. At low population, growth can be slow because there are few individuals.
In the middle range, growth is quickest. Near the limit, growth slows.
When using data, pay attention to units, starting time, and whether the rate constant is measured per day, per year, or per hour. A correct equation with mismatched units gives a misleading prediction.