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First-Order ODEs

Tertiary

Differential Equations

Concept

Separable, linear, and exact first-order ordinary differential equations.

Walkthrough

Solve dy/dx = 2x with y(0) = 1.

  1. 1.This is a separable ODE. Rearrange so all y terms are on one side and x terms on the other.
  2. 2.Separate: dy = 2x dx.
  3. 3.Integrate both sides: ∫dy = ∫2x dx, which gives y = x² + C.
  4. 4.Apply the initial condition y(0) = 1: 1 = (0)² + C, so C = 1.
  5. 5.So the solution is y = x² + 1.

Answer

y = x² + 1

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