Unit 2 of 13

First-Order Solution Methods

  • 7 lessons
  • Videos 7 to 13

Many first-order equations can be solved exactly once their structure is recognized. This unit covers separable, linear, exact, homogeneous, Bernoulli and Riccati equations and integrating factors.

Separable and linear equations come first, with care for equilibria lost by division and for the interval where a solution is valid. Exact equations are solved through a potential function, which the test on mixed partial derivatives guarantees locally and on any simply connected domain. Integrating factors that depend on one variable make some nonexact equations exact. Substitutions reduce homogeneous equations to separable ones, and Bernoulli equations and Riccati equations with a known solution to linear ones. Each method is checked for the solution branches it leaves out.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Separable and Linear Equations

  1. Separable Equations

    How to solve y' = a(t)b(y) by separation of variables, why the chain rule justifies it, how to keep the equilibria lost by division, and how to read an implicit solution and its solution interval.

    Video 7

  2. Linear First-Order Equations and Integrating Factors

    How to put a linear first-order equation in normalized form, why an integrating factor makes its left side the derivative of a product, and how continuous coefficients give the definite-integral solution formula.

    Video 8

Exact Equations and Integrating Factors

  1. Exact Differential Equations

    How to recognize M dx + N dy as an exact differential, how the test for exactness leads to a potential function for an exact equation, and why local exactness becomes global on a simply connected domain.

    Video 9

  2. Integrating Factors for Nonexact Equations

    How a multiplier that depends on x alone or on y alone can make a nonexact equation exact, how the two quotient tests find it, and which domain restrictions decide where the new equation is equivalent.

    Video 10

Substitution Methods

  1. Homogeneous First-Order Equations and Substitution

    What makes a first-order equation homogeneous in the sense of scale invariance, how the ratio substitution v = y/x reduces it to a separable equation, and how to restore the solution branches that the method excludes.

    Video 11

  2. Bernoulli Equations

    How the power substitution reduces a Bernoulli equation to a linear equation, why the exponents 0 and 1 are exceptional, and how to restore the zero solution and respect the domain of the power.

    Video 12

  3. Riccati Equations with a Known Particular Solution

    How a known particular solution reduces a Riccati equation to a linear equation for the correction through a reciprocal substitution, why the omitted branch is kept separately, and where poles appear.

    Video 13