🕹️ Practice arcade

⚡ Differential Equations Sprint

Euler's method, exponential growth/decay, initial value problems, separable equations, integrating factors, Newton's Law of Cooling — 120 seconds.

⚡ One-Step Method Sprint

Compute Heun's corrector step and RK4's first stage -- the arithmetic a numerical solver repeats every iteration, 90 seconds.

🔁 Laplace Transform Table Sprint

Transform t^n, then invert a shifted pole -- the table lookups a Laplace solve leans on again and again.

🧮 Eigenvalue Method Sprint

Write a linear system's characteristic equation, then count how many real solutions a complex eigenpair actually produces -- 90 seconds.

📐 Boundary-Value Problem Sprint

Solve a Robin condition for its missing quantity, then read a Dirichlet eigenvalue off n and L -- 90 seconds.

⚡ IVP, BVP, or Neither? Sprint

Classify a problem as an IVP or a BVP, then check whether the existence-uniqueness hypothesis actually holds -- sixty seconds.

⚡ Isoclines & Parameter Equilibria Sprint

Find an isocline's y-intercept, then track an equilibrium as a parameter moves it -- ninety seconds.

⚡ Mixing, Circuit & Pharmacokinetic Models Sprint

A two-tank inflow concentration, an RC circuit's time constant, a compartment's outflow-only rate, a drug's half-life -- four first-order models, ninety seconds.

⚡ Wronskian & Independence Sprint

Compute a Wronskian, check independence by ratio, or run Abel's formula forward -- ninety seconds.

⚡ Characteristic Root Types Sprint

Distinct real roots, a repeated root, or a complex pair's envelope frequency -- solve a constant-coefficient equation's roots three ways, ninety seconds.

⚡ Undetermined Coefficients Sprint

Look up a trial form, catch it when resonance forces a modification, count a superposed forcing term's own constants, or track variation of parameters' sign pattern -- ninety seconds.

⚡ Higher-Order Root Counting Sprint

Count an nth-order equation's independent solutions, find its characteristic polynomial's degree, or check that root multiplicities sum to the order -- ninety seconds.

⚡ Frobenius & Special Functions Sprint

Solve an indicial equation for its larger root, check that term-by-term calculus preserves the radius of convergence, catch the case that needs a log term, or match a Bessel/Legendre recurrence to its symmetry -- ninety seconds.

📐 Sturm-Liouville & Fourier Series Sprint

Find a Sturm-Liouville integrating factor, check weighted versus unweighted orthogonality, catch a Dirichlet rod's series type, or read a separated mode's decay rate -- ninety seconds.

🌀 Damping Regime Survival

Overdamped, critically damped, or underdamped -- misclassify the regime once and the round ends.

🧭 Jacobian and Hyperbolicity Survival

Build a Jacobian entry, then check whether Hartman-Grobman actually applies at that equilibrium -- one wrong call ends the round.

🧭 Bifurcation & Long-Term Behavior Survival

Find the bifurcation value of a one-parameter family, the harvested logistic equation's stable equilibrium, or where a solution ends up as t grows -- one wrong call ends the round.

🎯 Order and Stability

A differential equation's order, and whether an autonomous equilibrium is stable or not — five in a row.

🔑 Initial Value Mastery

Integrate, then use the initial condition to pin down $C$.

🎯 Exact or Homogeneous?

Check M_y against N_x for exactness, then see whether a first-order equation reduces to a function of y/x alone -- five in a row.

🎯 Nodes and Saddles

Read trace and determinant, classify the equilibrium -- node or saddle, five in a row.

🎯 Ordinary or Singular?

Classify a point as ordinary or singular, then regular-singular or irregular-singular -- the check that decides which series method applies, five in a row.

🎯 Linear or Nonlinear? Explicit or Implicit?

Classify an equation as linear or nonlinear, then a solution as explicit or implicit -- five in a row.

🎯 Which Model Fits?

Match a word problem to its differential equation, read off a reaction's order, combine a building's heating/cooling terms, or find a falling object's terminal velocity -- five in a row.

🎯 Standard Form, Homogeneity & Superposition

Divide into standard form, tell a homogeneous equation from a forced one, and check that superposition actually applies -- five in a row.

🎯 Fundamental Sets & General Solutions

Check whether a pair of solutions forms a fundamental set, build the general solution from an IVP's data, or find reduction of order's second solution -- five in a row.

🎯 Cauchy-Euler & Long-Run Behavior

Solve a Cauchy-Euler equation's roots, handle its repeated-root case, then read a solution's long-run behavior straight off the characteristic roots -- five in a row.

🎯 Forced Vibration Response

Pick out the steady-state term, compute an amplitude response, find a phase shift's quadrant, or read a beat pattern's envelope frequency -- five in a row.

🎯 Converting to a First-Order System

Count how many first-order equations an nth-order conversion produces, relabel the right number of variables, and size a variation-of-parameters system -- five in a row.

🎯 Numerical Method Theory

Read local error against global error, back out a method's order from an error ratio, or find a stiff equation's stability threshold -- five in a row, no arithmetic required.

🎯 Equilibria, Stability & Trajectories

Find a linear system's only equilibrium, read stability off the trace alone, count a trajectory's dimension, or read a direction field's arrow -- five in a row.

🎯 Centers, Spirals & Degenerate Cases

Check the center condition, read a spiral's stability off the sign of the trace, or work out what a zero determinant means for the equilibrium -- five in a row.

🎯 Verifying Systems Solutions

Check that a vector solves a system, spot a repeated eigenvalue's second-solution shape, keep a decoupled system's equations apart, or catch the matrix exponential's own derivative -- five in a row.