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Differential Equations — ODEs

Solving and classifying ordinary differential equations

D
diffeq_drifter 25 terms Feb 28, 2026
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Terms 25

1
Ordinary Differential Equation (ODE)
Equation relating a function with its derivatives; involves one independent variable
2
Order of ODE
Highest derivative present in the equation
3
Separable ODE
dy/dx = f(x)g(y); separate variables and integrate both sides
4
Linear First-Order ODE
dy/dx + P(x)y = Q(x); solved using integrating factor μ = e^∫P(x)dx
5
Integrating Factor
μ(x) = e^∫P(x)dx; multiply through to make left side exact derivative
6
Exact ODE
M(x,y)dx + N(x,y)dy = 0 where ∂M/∂y = ∂N/∂x; solved by finding potential function
7
Homogeneous ODE (linear)
ay'' + by' + cy = 0; solution depends on characteristic equation ar² + br + c = 0
8
Characteristic Equation
ar² + br + c = 0; roots determine form of general solution to homogeneous ODE
9
Case 1: Real Distinct Roots
r₁ ≠ r₂; general solution: y = C₁e^(r₁x) + C₂e^(r₂x)
10
Case 2: Repeated Root
r₁ = r₂ = r; general solution: y = (C₁ + C₂x)e^(rx)
11
Case 3: Complex Roots
r = α ± βi; general solution: y = e^(αx)(C₁cos(βx) + C₂sin(βx))
12
Method of Undetermined Coefficients
Guesses particular solution form for non-homogeneous ODE based on right-hand side
13
Variation of Parameters
More general method for particular solution; works for any continuous right-hand side
14
Laplace Transform
L{f(t)} = ∫₀^∞ e^(−st)f(t)dt; converts ODE to algebraic equation
15
Inverse Laplace Transform
Recovers f(t) from F(s); uses partial fractions and transform tables
16
Initial Value Problem (IVP)
ODE with conditions specifying function value (and derivatives) at a starting point
17
Euler's Method
Numerical approximation: y_{n+1} = y_n + h·f(x_n, y_n); tangent line stepping
18
Phase Plane
Plot of y' vs y; visualizes solutions of autonomous systems without solving explicitly
19
Equilibrium Point
Constant solution where dy/dt = 0; stable (attracting) or unstable (repelling)
20
Linearization
Approximating nonlinear system near equilibrium using Jacobian matrix; determines stability
21
Systems of ODEs
Vector equation x' = Ax; general solution uses eigenvalues and eigenvectors of A
22
Bifurcation
Qualitative change in behavior of system as parameter varies; e.g. stable equilibrium becomes unstable
23
Logistic Equation
dP/dt = rP(1 − P/K); models population growth with carrying capacity K
24
Bernoulli Equation
dy/dx + P(x)y = Q(x)yⁿ; substitution v = y^(1−n) linearizes it
25
Fourier Series
Represents periodic function as sum of sines and cosines; used to solve PDEs