Complete mastery of ordinary differential equations and their applications
← Back to MathematicsUnderstand the fundamental concepts and terminology of differential equations.
Master the technique of separation of variables for first-order equations.
Solve first-order linear equations using integrating factors.
Learn to solve exact equations and other special first-order methods.
Apply first-order differential equations to real-world modeling problems.
Understand the theory and general solutions of second-order linear equations.
Solve second-order equations with constant coefficients using characteristic equations.
Find particular solutions to nonhomogeneous equations with specific forcing functions.
Use variation of parameters for nonhomogeneous equations with any forcing function.
Model mechanical and electrical systems using second-order differential equations.
Use Laplace transforms to solve differential equations with discontinuous forcing.
Solve systems of first-order differential equations using matrix methods.
Build foundational understanding of differential equations, their classification, and basic concepts.
Find integrating factors when equations are not exact, making them exact for solution. Learn systematic approaches for finding μ(x) or μ(y).
Solve homogeneous equations dy/dx = f(y/x) using the substitution v = y/x. Transform to separable form and solve.
Use various substitutions to transform difficult equations into solvable forms. Learn when and how to apply different substitution techniques.
Identify equations that can be reduced to separable form through appropriate substitutions. Master systematic approaches to equation transformation.
Apply specialized transformations and substitutions for specific equation types, including equations of the form dy/dx = f(ax + by + c).
Apply first-order differential equations to model and solve real-world problems across various fields.
Learn the systematic approach to mathematical modeling: problem identification, assumption making, equation formulation, solving, and interpretation.
Model various population scenarios including exponential growth, logistic growth with carrying capacity, and populations with harvesting or immigration.
Understand basic predator-prey interactions using first-order equations. Model competition and cooperation between species.
Apply differential equations to economics: supply and demand dynamics, price equilibrium, and market behavior over time.
Model falling objects with air resistance, terminal velocity problems, and motion in viscous media using first-order equations.
Model chemical reaction rates, first-order reactions, and concentration changes over time using differential equations.
Analyze RC and RL circuits, charging and discharging capacitors, and current behavior in simple electrical systems.
Find families of curves that intersect given curve families at right angles. Applications to heat flow and electric field lines.
Build theoretical foundation for second-order linear differential equations and their solution structure.
Understand the general form a(x)y″ + b(x)y′ + c(x)y = f(x) and distinguish between homogeneous and nonhomogeneous equations.
Distinguish between homogeneous equations (f(x) = 0) and nonhomogeneous equations (f(x) ≠ 0). Understand solution structure differences.
Understand linear independence of functions and its importance in forming general solutions. Learn to test independence using various methods.
Calculate and use the Wronskian W(y₁,y₂) = y₁y₂′ - y₂y₁′ to test linear independence of solutions.
Identify fundamental sets of solutions for homogeneous equations. Understand how any solution can be expressed as their linear combination.
Learn that general solutions to nonhomogeneous equations have the form y = y_h + y_p, where y_h is homogeneous and y_p is particular.
Solve initial value problems for second-order equations with conditions y(x₀) = y₀ and y′(x₀) = y₁. Apply initial conditions to find constants.
Apply the superposition principle: if y₁ and y₂ are solutions to a linear homogeneous equation, then c₁y₁ + c₂y₂ is also a solution.
Master the characteristic equation method for solving second-order equations with constant coefficients.
Solve equations of the form ay″ + by′ + cy = 0 where a, b, c are constants. Learn the systematic approach using characteristic equations.
Form and solve the characteristic equation ar² + br + c = 0 by assuming solutions of the form y = e^(rx). Apply the quadratic formula.
Handle the case when the characteristic equation has two real, distinct roots r₁ and r₂. General solution: y = c₁e^(r₁x) + c₂e^(r₂x).
Solve cases with repeated roots (r₁ = r₂ = r). General solution: y = (c₁ + c₂x)e^(rx). Understand why the x factor appears.
Handle complex roots r = α ± βi. Use Euler's formula to find real-valued solutions involving sine and cosine functions.
Apply Euler's formula e^(iθ) = cos(θ) + i sin(θ) to convert complex exponential solutions to real trigonometric solutions.
Convert complex solutions to real form: y = e^(αx)(c₁cos(βx) + c₂sin(βx)) when roots are α ± βi.
Apply constant coefficient equations to model spring-mass systems, pendulums, and other mechanical oscillators.
Learn to find particular solutions for nonhomogeneous equations with polynomial, exponential, and trigonometric forcing functions.
Understand the structure ay″ + by′ + cy = f(x) where f(x) ≠ 0. Learn that solutions combine homogeneous and particular parts.
Learn the systematic approach: guess the form of particular solution based on f(x), substitute into equation, solve for coefficients.
Handle forcing functions f(x) = polynomial. Guess particular solutions as polynomials of the same or higher degree.
Solve equations with f(x) = Ae^(kx). Guess y_p = Be^(kx) and determine coefficient B by substitution.
Handle f(x) involving sin(kx) or cos(kx). Guess combinations of both sine and cosine terms even if only one appears in f(x).
Deal with forcing functions that are products (like xe^x) or sums of basic functions. Apply linearity and product rules for guessing.
Handle cases where the guessed particular solution is also a solution to the homogeneous equation. Multiply by x or x² as needed.
Combine homogeneous and particular solutions, then apply initial conditions to find the complete solution to initial value problems.
Master the variation of parameters method for finding particular solutions to any nonhomogeneous linear equation.
Learn the variation of parameters approach: assume particular solution y_p = u₁y₁ + u₂y₂ where u₁, u₂ are functions to be determined.
Derive the system of equations for u₁′ and u₂′, then integrate to find u₁ and u₂. Construct the particular solution.
Use the Wronskian W(y₁,y₂) in the formulas: u₁′ = -y₂f(x)/W and u₂′ = y₁f(x)/W where f(x) is the forcing function.
Master the systematic steps: find homogeneous solutions, compute Wronskian, apply variation formulas, integrate to find u₁ and u₂.
Understand when to use each method: undetermined coefficients for specific f(x) types, variation of parameters for any f(x).
Apply variation of parameters to equations with variable coefficients, where undetermined coefficients cannot be used.
Extend the method to third-order and higher-order linear equations. Understand the general pattern for n-th order equations.
Apply variation of parameters to analyze forced mechanical and electrical oscillations with arbitrary driving functions.
Model and analyze mechanical and electrical systems using second-order differential equations.
Model spring-mass systems using Hooke's law and Newton's second law: mx″ + kx = 0. Understand natural frequency and period.
Analyze undamped free oscillations (simple harmonic motion) and damped oscillations with friction or air resistance.
Study forced oscillations mx″ + cx′ + kx = F(t) with external driving forces. Understand steady-state and transient responses.
Model RLC electrical circuits using Kirchhoff's laws: LI″ + RI′ + I/C = E′(t). Analyze current and voltage behavior.
Classify damping: underdamped (oscillatory), critically damped (fastest return to equilibrium), and overdamped (slow return).
Understand resonance when driving frequency matches natural frequency. Analyze amplitude response and phase relationships.
Study beats phenomenon when driving frequency is close to natural frequency. Understand amplitude modulation effects.
Introduce nonlinear effects in pendulum motion. Compare small-angle approximation with exact nonlinear equation behavior.
Use Laplace transforms to solve differential equations with discontinuous forcing functions and initial conditions.
Understand the Laplace transform L{f(t)} = ∫₀^∞ e^(-st)f(t)dt. Learn when transforms exist and their basic properties.
Memorize and derive Laplace transforms of basic functions: constants, powers, exponentials, trigonometric, and hyperbolic functions.
Master linearity, first and second shifting theorems, scaling, and transforms of derivatives: L{f′} = sL{f} - f(0).
Find inverse Laplace transforms using tables, partial fractions, and completing the square. Understand uniqueness of inverse transforms.
Apply Laplace transforms to solve initial value problems: transform equation, solve algebraically, then inverse transform.
Use the unit step function u(t-a) to model discontinuous forces. Apply the second shifting theorem for delayed functions.
Understand the Dirac delta function δ(t-a) for modeling impulses. Learn L{δ(t-a)} = e^(-as) and impulse response applications.
Apply the convolution theorem: L^(-1){F(s)G(s)} = (f*g)(t) where (f*g)(t) = ∫₀^t f(τ)g(t-τ)dτ. Find transfer functions.
Solve systems of first-order linear differential equations using matrix methods and eigenvalue techniques.
Convert higher-order equations and coupled systems into first-order systems. Write in matrix form x′ = Ax + b.
Express systems as x′ = Ax where x is the vector of unknown functions and A is the coefficient matrix. Understand matrix notation.
Find eigenvalues and eigenvectors of coefficient matrix A. Use them to construct fundamental matrix and general solution.
Handle cases with real, distinct eigenvalues λ₁, λ₂. General solution: x = c₁v₁e^(λ₁t) + c₂v₂e^(λ₂t).
Deal with complex eigenvalues λ = α ± βi. Use Euler's formula to find real-valued solutions involving sine and cosine.
Handle repeated eigenvalues with insufficient eigenvectors. Find generalized eigenvectors and use Jordan canonical form methods.
Sketch phase portraits showing solution trajectories in the phase plane. Classify equilibrium points: nodes, spirals, saddles.
Determine stability of equilibrium points based on eigenvalues. Understand asymptotic stability, instability, and neutrally stable cases.