Differential Equations Formula Sheet

Differential equations formula sheet

This sheet is the ODE companion to the Phase Portrait Generator: first- and second-order equations, linearization, and the eigenvalue test that classifies a node, saddle, spiral, or center. It does not replace the plotter—use the generator to draw the field, then check the classification against the Jacobian here.

First-order and linear second-order

separable:     dy/dx = g(x) h(y)  →  ∫ dy/h(y) = ∫ g(x) dx
linear 1st:    y′ + P(x) y = Q(x)
  μ = exp(∫ P dx),   (μ y)′ = μ Q
constant-coeff:  y″ + b y′ + c y = 0
  characteristic r² + b r + c = 0
  two real r:     y = A e^{r₁ t} + B e^{r₂ t}
  repeated r:     y = (A + B t) e^{r t}
  complex α±iβ:   y = e^{α t} (A cos βt + B sin βt)

2D autonomous systems

ẋ = f(x, y)
ẏ = g(x, y)

equilibrium (x*, y*):  f = 0 and g = 0
nullclines:            f = 0  (vertical flow),  g = 0  (horizontal flow)

Trajectories are tangent to the vector field and do not cross (uniqueness). Click the phase plane in the Phase Portrait Generator to integrate with RK4. The Center, Spiral sink, and Saddle presets match the linear cases below.

Jacobian and classification

J = | ∂f/∂x  ∂f/∂y |
    | ∂g/∂x  ∂g/∂y |     evaluated at (x*, y*)

Eigenvalues λ of J:

  two real, same sign     node (sink if λ < 0, source if λ > 0)
  two real, opposite sign saddle (unstable)
  λ = α ± iβ, α ≠ 0       spiral (sink if α < 0, source if α > 0)
  λ = ± iβ                center (closed orbits, neutrally stable)
  one λ = 0               non-isolated or degenerate — linear test fails

The linear solver helps when you already have J x = λ x as a matrix problem: Linear Equations Solver.

Worked linear portraits

center:   ẋ = −y,  ẏ = x           J = [[0,-1],[1,0]]   λ = ±i
saddle:   ẋ = x,   ẏ = −y          λ = +1, −1
sink:     ẋ = −x,  ẏ = −2y         λ = −1, −2
spiral:   ẋ = −0.3x − y, ẏ = x − 0.3y    λ = −0.3 ± i

Load those as presets on the plotter, then enable equilibria + classification. Nonlinear examples (Van der Pol, damped pendulum) have the same local types after linearization, plus possible limit cycles that a linear J cannot see.

Existence, uniqueness, and time

Lipschitz f,g  → unique solution through each (x₀, y₀)
non-autonomous ẋ = f(x, t) is not a 2D phase portrait
  (use the time-graphing tool for x(t) instead)

For explicit y = f(x) plots use the 2D Graphing Calculator; for x(t) use the Time Graphing Tool.