Separable equation
$$\frac{dy}{dx}=g(x)h(y)\implies\int\frac{dy}{h(y)}=\int g(x)dx$$
Separate variables so each side can be integrated.
UniversityDifferential equations formulas, grouped by topic.
8 differential equations formulas across 5 topics, free and without an account. Differential-equation formulas solve for a function from a statement about its own rate of change. Separable and linear first-order methods handle most introductory models, characteristic roots produce the general solution of a constant-coefficient second-order equation, and Laplace transforms convert an initial-value problem into algebra. Every general solution needs initial conditions before it describes one particular situation.
Every differential equations formula on one page
Each entry links to its own page, where the symbols are defined, the conditions are stated, and the formula is substituted through a worked example. Levels covered here: AP, University.
First order
2 formulas in first order.
Linear integrating factor
$$y'+p(x)y=q(x),\quad\mu(x)=e^{\int p(x)dx}$$
Turn a first-order linear equation into one product derivative.
UniversityModels
1 formula in models.
Logistic growth
$$\frac{dP}{dt}=rP\left(1-\frac PK\right),\qquad P(t)=\frac{K}{1+Ae^{-rt}}$$
Model growth that slows near a carrying capacity.
AP · UniversitySecond order
2 formulas in second order.
Constant-coefficient characteristic equation
$$ay''+by'+cy=0\implies ar^2+br+c=0$$
Turn a homogeneous linear differential equation into an algebraic root problem.
UniversityHomogeneous solution forms
$$r_1\ne r_2:\ y=C_1e^{r_1x}+C_2e^{r_2x};\quad r:\ y=(C_1+C_2x)e^{rx};\quad \alpha\pm i\beta:\ y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)$$
Write the real general solution from characteristic roots.
UniversityNumerical methods
2 formulas in numerical methods.
Euler’s method
$$y_{n+1}=y_n+h f(x_n,y_n),\qquad x_{n+1}=x_n+h$$
Step along a differential equation using the current tangent slope.
AP · UniversityFourth-order Runge–Kutta
$$y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4)$$
Advance an initial-value solution using four slope samples.
UniversityLaplace transforms
1 formula in laplace transforms.
Derivative transform
$$\mathcal L\{f'(t)\}=sF(s)-f(0)$$
Convert a time derivative into algebra while preserving an initial value.
UniversityKeep going with differential equations
A formula sheet is a reference, not a method. Use the subject guide to see where these relationships come from, a calculator to check a substitution you are unsure about, and practice questions to find out whether you can actually retrieve them.
- Study differential equations — the subject hub with lessons, worked methods, and common mistakes.
- Calculus calculator — enter your own values and compare the result with your working.
- Differential Equations Practice — retrieval practice across every topic listed above.
- Math formula library — search all 215 formulas by name, symbol, or condition.
Formulas in the other subjects
Mathematics does not stop at a subject boundary, and neither do its formulas. Each subject page below lists its own relationships, conditions, and worked examples.
Arithmetic
Review the core arithmetic relationships, their restrictions, and worked substitutions.
Algebra
Review the core algebra relationships, their restrictions, and worked substitutions.
Geometry
Review the core geometry relationships, their restrictions, and worked substitutions.
Trigonometry
Review the core trigonometry relationships, their restrictions, and worked substitutions.
Precalculus
Review the core precalculus relationships, their restrictions, and worked substitutions.
Calculus
Review the core calculus relationships, their restrictions, and worked substitutions.
Statistics
Review the core statistics relationships, their restrictions, and worked substitutions.
Probability
Review the core probability relationships, their restrictions, and worked substitutions.
Linear Algebra
Review the core linear algebra relationships, their restrictions, and worked substitutions.
Discrete Math
Review the core discrete math relationships, their restrictions, and worked substitutions.