Source: MasterMathMentor Ch. 39–40, Advanced Calculus I (UCF)
Tags: slope fields, direction fields, differential equations, separable DEQ, general solution, particular solution, initial condition, separation of variables, dy/dx, antiderivative
Difficulty: Intermediate to Advanced Prerequisites: All basic integration techniques (power rule, substitution, trig integrals, exponential/log integrals), the concept of a derivative as slope, and comfort with implicit equations. Chapter 25 (simple differential equations) is the direct precursor.
A differential equation (DEQ) relates a function to its own derivative. Up to now, you have been told a function and asked to differentiate it, or told a derivative and asked to integrate it. Differential equations flip the perspective: you are given a relationship involving dy/dx and asked to find the original function y. Slope fields give you a visual way to see all possible solutions at once, and separation of variables is the algebraic technique that lets you solve a large, important class of these equations exactly. These two chapters together form the gateway to differential equations, a subject you will study extensively in college-level maths and engineering.
A slope field draws tiny line segments showing the slope dy/dx at each point, revealing the shape of solution curves without solving anything. A separable differential equation is one where you can move all the y-stuff to one side and all the x-stuff to the other, then integrate both sides to find the solution. Plugging in an initial condition pins down the constant C and gives the particular solution.
Differential equation (DEQ)
An equation involving a function and one or more of its derivatives. In this course, it takes the form dy/dx = some expression involving x and/or y. Think of it as a recipe that tells you the slope everywhere, and your job is to find the curve that follows that recipe.
Slope field (direction field)
A grid of short line segments, each drawn at a point (x, y) with the slope given by the DEQ at that point. It is a visual map of all possible solution curves. In simple terms, it is like drawing tiny arrows showing which way the curve is heading at every location.
General solution
The family of all solutions to a DEQ, expressed with an arbitrary constant C. Every value of C gives a different curve, and together they tile the slope field.
Particular solution
The single solution curve that passes through a specified point (x_0, y_0). Found by substituting the initial condition into the general solution to determine C.
Initial condition
A given point (x_0, y_0) that the solution must pass through. It selects one curve from the infinite family of general solutions.
Separable differential equation
A DEQ that can be rewritten in the form g(y) dy = f(x) dx, with all y-terms (including dy) on one side and all x-terms (including dx) on the other. This allows you to integrate each side independently.
Non-separable differential equation
A DEQ where the variables cannot be cleanly separated. Examples: dy/dx = x + y, dy/dx = (y - x)/(x + 2y). These require other techniques not covered in this course.
How to build one:
Pick a grid of points, typically integers from -2 to 2 in both x and y.
At each point, substitute (x, y) into the DEQ to compute dy/dx.
Draw a short line segment with that slope at that point.
Where dy/dx = 0, draw a horizontal segment. Where dy/dx is undefined, draw a vertical segment or leave it blank.
How to read one:
Visually trace a curve that is tangent to every segment it passes through. That curve is an approximate solution.
If given an initial condition, start at that point and follow the field in both directions.
The shape of the solution curves often reveals the type of function: parabolas, exponentials, logistic curves, etc.
Pattern recognition tips:
If dy/dx depends only on y (e.g. dy/dx = y), slopes are the same along every horizontal line. All segments in the same row look identical.
If dy/dx depends only on x (e.g. dy/dx = 2x), slopes are the same along every vertical column.
If dy/dx depends on both x and y, slopes vary across both rows and columns.
The method, step by step:
Confirm the DEQ is separable by checking whether you can factor or rearrange it into g(y) dy = f(x) dx.
Move all y-terms to the left with dy, all x-terms to the right with dx.
Integrate both sides. Place the constant of integration C on one side only (conventionally the right).
Solve for y if practical. If solving for y explicitly is messy, leaving the answer in implicit form is acceptable.
This result is the general solution.
Recognising separable form:
dy/dx = xy is separable: (1/y) dy = x dx.
dy/dx = y^2 is separable: (1/y^2) dy = dx.
dy/dx = (sin t)/(cos y) is separable: cos y dy = sin t dt.
dy/dx = x + y is NOT separable: you cannot isolate all y-terms from all x-terms.
dy/dx = (y - x)/(x + 2y) is NOT separable.
dy/dx = y - e^(-x) is NOT separable.
Finding the particular solution:
After finding the general solution (with C), substitute the initial condition (x_0, y_0) to solve for C.
Replace C with the numerical value to write the particular solution.
At every point, the slope equals the y-coordinate. Row y = 2 has slope 2 everywhere. Row y = 0 has slope 0 everywhere (horizontal segments). Row y = -1 has slope -1 everywhere. The solution curves are exponentials. Through (1, 1), the particular solution is y = e^(x-1).
Separate: y dy = 2x dx. Integrate: y^2/2 = x^2 + C, or y^2 = 2x^2 + C. General solution: y = plus or minus sqrt(2x^2 + C).
Separate: y dy = (1 + x)/x dx = (1/x + 1) dx. Integrate: y^2/2 = ln|x| + x + C. Apply y(1) = -4: 16/2 = ln(1) + 1 + C, so 8 = 0 + 1 + C, giving C = 7. Particular solution: y^2/2 = ln|x| + x + 7, or y^2 = 2 ln|x| + 2x + 14.
Separate: (1/y) dy = 4 dx. Integrate: ln|y| = 4x + C. Exponentiate: y = Ae^(4x) where A = plus or minus e^C. Verification: dy/dx = 4Ae^(4x) = 4y. Confirmed.
Separation of variables template: g(y) dy = f(x) dx, then integral of g(y) dy = integral of f(x) dx + C.
Common results to memorise:
dy/dx = ky has general solution y = Ce^(kx).
dy/dx = xy has general solution y = Ce^(x^2/2).
dy/dx = y^2 has general solution y = -1/(x + C).
Separable DEQs model any situation where the rate of change of a quantity depends on the quantity itself (or can be separated into independent parts). Newton's Law of Cooling, population growth, radioactive decay, and compound interest all reduce to separable equations. The next two chapters (41 and 42) are entirely applications of these techniques.
Students often try to separate dy/dx = x + y by moving y to the left. This does not work because you cannot write it as g(y) dy = f(x) dx. The y is added, not multiplied.
Forgetting the constant C after integration is a classic error. Without C, you have only one solution curve instead of the entire family.
When exponentiating ln|y| = kx + C, students sometimes write y = e^(kx) + e^C instead of y = e^C e^(kx). The exponent rule is e^(a+b) = e^a e^b (multiplication, not addition).
Students occasionally drop the absolute value when integrating 1/y. This matters when y can be negative: the constant C absorbs the sign, but you need to be aware of it.
⚠️ AP Calculus exams regularly include a slope-field matching question in the multiple-choice section. Practice recognising whether dy/dx depends on x only, y only, or both.
⚠️ Free-response questions often ask you to solve a separable DEQ and find the particular solution. Every step must be shown: separate, integrate, apply initial condition, solve for y.
⚠️ You may be asked to sketch a solution curve on a given slope field. Use a smooth curve that follows the segments, starting from the given point.
⚠️ If the question says "find the general solution," leave C in. If it says "find the particular solution," you must determine C from the initial condition.
True or false: dy/dx = x^2 * y is a separable differential equation.
Fill in the blank: a slope field shows the value of ______ at each point (x, y).
True or false: the general solution to dy/dx = ky is y = ke^(Cx).
If dy/dx depends only on y, all segments in the same ______ of the slope field have the same slope.
True or false: when solving a separable DEQ, you put a constant C on both sides of the equation after integrating.
Answers: 1) True (separate as (1/y) dy = x^2 dx). 2) dy/dx (the slope). 3) False (it is y = Ce^(kx)). 4) Row (same y-value). 5) False (one constant on one side is sufficient, since the difference of two constants is still a constant).
Q: Is dy/dx = (y - x)/(x + 2y) separable? Why or why not?
A: No. You cannot factor the right side into a product of a function of x alone and a function of y alone. The x and y are entangled in both the numerator and denominator.
Q: Solve dy/dx = x/y and find the particular solution through (0, 3).
A: Separate: y dy = x dx. Integrate: y^2/2 = x^2/2 + C. Apply (0, 3): 9/2 = 0 + C, so C = 9/2. Solution: y^2 = x^2 + 9, or y = sqrt(x^2 + 9) (taking the positive root since y(0) = 3 > 0).
Q: For the slope field of dy/dx = x + y, what is the slope at the point (1, -1)?
A: dy/dx = 1 + (-1) = 0. The segment at (1, -1) is horizontal.
Q: Solve dy/dx = e^(y + 2t) with respect to t.
A: Rewrite as dy/dx = e^y * e^(2t). Separate: e^(-y) dy = e^(2t) dt. Integrate: -e^(-y) = (1/2)e^(2t) + C. Solve: e^(-y) = -(1/2)e^(2t) - C, so -y = ln(-(1/2)e^(2t) - C), giving y = -ln(-(1/2)e^(2t) - C).
Slope fields connect to the graphical interpretation of derivatives from early in the course: the derivative is slope, and the slope field simply shows that slope everywhere at once. Separable DEQs connect directly to all integration techniques you have learned (substitution, partial fractions, trig integrals) since those are the tools you use after separating. Chapters 41 and 42 are entirely built on the separable DEQ dy/dx = kP and its variants.
slope field calculus, direction field, separable differential equation, separation of variables, general solution DEQ, particular solution initial condition, dy/dx = ky solution, AP Calculus slope field matching, exponential DEQ, implicit solution differential equation, constant of integration, solving DEQ step by step