In Differential Equations, 80% of the battle is classifying the equation on sight.
On homework, you know which section the problem came from. On the midterm, all equation types are scrambled together. Here is your classification decision tree.
The 5-step Differential Equations classification & solving system
Classify First-Order ODEs in strict priority order
Check in this exact order: (1) Is it Separable (dy/dx = g(x)h(y))? (2) Is it Linear (dy/dx + P(x)y = Q(x) → use Integrating Factor μ(x) = e^∫P(x)dx)? (3) Is it Exact (∂M/∂y = ∂N/∂x)? (4) Is it Bernoulli or Homogeneous?
Solve 2nd-Order Homogeneous ODEs via the Characteristic Equation
Turn ay'' + by' + cy = 0 into ar² + br + c = 0. Distinct real roots give c₁e^(r₁t) + c₂e^(r₂t); repeated root gives (c₁ + c₂t)e^(rt); complex roots α ± βi give e^(αt)(c₁cos(βt) + c₂sin(βt)).
Avoid the "Resonance Overlap" trap in Undetermined Coefficients
When guessing the particular solution y_p(t) for ay'' + by' + cy = g(t), ALWAYS check your homogeneous solution y_h(t) first. If your guess duplicates a term in y_h, multiply your guess by t (or t²)!
Use Laplace Transforms to turn differential equations into algebra
Take L{y''} = s²Y(s) - sy(0) - y'(0), solve algebraically for Y(s), use partial fractions + completing the square, and take the inverse Laplace transform L⁻¹{Y(s)}.
Verify every solution in 30 seconds by plugging y and y' back in
Differential Equations is the easiest math class to check your own work: differentiate your answer and verify that it satisfies both the ODE and the initial conditions y(0).
Core ODE exam topics to drill with AI
1st-Order Classification Drills
Practice identifying Separable vs. Integrating Factor vs. Exact ODEs without solving.
Spring-Mass & RLC Oscillators
Model underdamped, critically damped, and overdamped motion plus resonance.
Laplace, Step & Dirac Delta Functions
Handle piecewise forcing functions u_c(t) and impulse hammers δ(t - t₀).
Systems of ODEs & Phase Portraits
Use eigenvalues of X' = AX to classify sinks, sources, saddle points, and spirals.
Put 1st-order linear ODEs into Standard Form before finding P(x)
If the equation is x(dy/dx) + 2y = x³, you MUST divide every term by x first to get dy/dx + (2/x)y = x² before computing the integrating factor μ(x) = e^∫(2/x)dx = x².
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