Try These Examples
Calculus
Find the integral of x²·eˣ dx
Physics
Maximum height of a ball thrown upward at 20 m/s
Chemistry
Balance: Fe + O₂ → Fe₂O₃
Algebra
Solve: 2x + 3y = 12, x - y = 1
Biology
Find ORFs, transcribe, translate, and GC content of a DNA sequence
Quantum
Bell state entanglement and von Neumann entropy
Simulation
Lotka-Volterra predator-prey dynamics simulation
Calculus
Solve y'' + 4y' + 4y = e⁻²ˣ·ln(x) via variation of parameters
Physics
Carnot engine: work, efficiency, and entropy between 800K and 300K
Chemistry
Nernst equation: Zn-Cu galvanic cell potential at non-standard conditions
Genomics
Analyze DNA: ORFs, restriction sites, GC profile, CpG islands
Biology
Hardy-Weinberg selection: allele frequency shift over 10 generations
Simulation
SIR epidemic model: peak timing and total infection fraction
Quantum
Spin-1/2 time evolution under combined σₓ + σ_z Hamiltonian
Statistics
One-sample t-test: is sample mean 52.3 different from population mean 50?
Linear Algebra
Eigendecomposition of 3×3 matrix, diagonalizability, and A¹⁰
Diff. Equations
Stability analysis of nonlinear 2D system with equilibrium classification
Signal Processing
4th-order Butterworth bandpass filter design with frequency response
Fluid Dynamics
Pipe flow: Reynolds number, flow regime, and Darcy-Weisbach pressure drop
Materials Science
Kronig-Penney band structure of 1D crystal: first 3 energy bands
Astrophysics
LEO satellite at 400 km: orbital period, velocity, energy, and escape velocity
Graph Theory
Dijkstra shortest path A→F and centrality measures in weighted graph
Stochastic
Black-Scholes European call pricing with Greeks (delta, gamma, vega)
Relativity
Muon at 0.99c: Lorentz factor, time dilation, length contraction, momentum