Subject GRE-Style: Solving a Separable ODE

GRE Math Subject · Axiom Academy

EXAMPLE GRE-Style: Solving First-Order ODEs Four worked problems covering separable, exact, and integrating factor techniques Solve the initial value problem , with . Speed check: Plug into each answer choice. Only (A) and (E) give . Then check (E): gives -- same! But at gives , and differentiating the original integral gives the same ODE. Checking more carefully: from , we get , confirming (A). Find the general solution of . Solution: Already in standard form with . Verification shortcut: The homogeneous solution of is . This eliminates (C). Among the rest, substitute the particular part into the ODE: , so . Solution: First verify exactness. With and : Equal, so the equation is exact. Find with and . Exam shortcut: Scan the answer choices. Any correct implicit solution must satisfy and . Differentiate each candidate to check -- often faster than solving from scratch. Solution: This is a Bernoulli equation with . Substitute . This is first-order linear in . Integrating factor: . Answer: None of the choices exactly match (the exponent should be , not ). On the actual GRE, verify your algebra carefully -- distractor choices often differ by a factor in the exponent. Bernoulli technique summary: For , substitute to reduce to a first-order linear ODE. This works for any . The GRE tests this roughly once every 2-3 exams.

This is the written version of the interactive lesson above. See the full GRE Math Subject course.