Differential Equations Summary

GRE Math Subject · Axiom Academy

SUMMARY Differential Equations Complete unit review: First-order, second-order, systems, and Laplace transforms Differential equations appear on roughly 10-15% of the GRE Math Subject Test. The emphasis is on recognizing ODE types and applying the correct solution method quickly. You rarely need to solve a full problem from scratch; instead, the GRE tests whether you can classify, set up, and identify the correct form of the solution. Form: . Separate variables and integrate both sides. GRE tip: Always check if separable first. It is the simplest method and applies more often than you think. Form: . Solve using the integrating factor . Key point: Get the equation into standard form (coefficient of must be 1) before identifying . Solution: Find where and . Then is the implicit solution. GRE shortcut: The exactness test is fast. If it fails, the equation might become exact with an integrating factor, but the GRE rarely goes there. Bernoulli: . Substitute to reduce to linear. Homogeneous: . Substitute to make separable. Form: . The characteristic equation determines the solution: GRE speed: The characteristic equation is a quadratic. Solve it, identify the case, write the answer. For where is a polynomial, exponential, sine/cosine, or combination. Guess the form of the particular solution based on : Modification rule: If your guess is already a solution to the homogeneous equation, multiply by (or for repeated roots).

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