Unit 4 of 13

Second-Order and Higher-Order Equations

  • 8 lessons
  • Videos 19 to 26

Linear equations of higher order have a clear solution structure. This unit builds it from superposition and the Wronskian, then solves equations with constant coefficients, forcing and special forms.

Where the normalized coefficients are continuous, the solutions of a homogeneous linear equation are the combinations of a fundamental set, and the Wronskian and Abel's identity test independence. Constant coefficients lead to the characteristic equation, whose roots can be distinct, repeated or complex. Undetermined coefficients or variation of parameters give a particular solution of a forced equation. Reduction of order finds a second solution from a known one, Cauchy-Euler equations become constant-coefficient equations in the logarithm of x, and some nonlinear equations drop an order.

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.

Linear Theory and Constant Coefficients

  1. Linear Equations, Superposition, and Wronskians

    How the linear differential operator leads to superposition, why two independent solutions form a fundamental set on a regular interval, and how the Wronskian and Abel's identity test independence.

    Video 19

  2. Second-Order Constant-Coefficient Equations

    How the characteristic equation solves a real constant-coefficient homogeneous equation of second order, with distinct real roots, a repeated root or complex conjugate roots, and how to keep the solutions real-valued.

    Video 20

  3. Higher-Order Constant-Coefficient Equations

    How the characteristic polynomial of an nth-order constant-coefficient equation and the multiplicity of its roots give n polynomial-exponential solutions, also for repeated complex roots, and how n initial values select one.

    Video 21

Particular Solutions of Forced Equations

  1. Undetermined Coefficients

    How the method of undetermined coefficients finds a particular solution from a trial function, why overlap multiplicity adds powers of t, and how annihilators explain the trial family.

    Video 22

  2. Variation of Parameters

    How variation of parameters builds a particular solution from a fundamental pair, why the Wronskian is the denominator, and how definite integrals give the response that starts at rest.

    Video 23

Reduction of Order and Cauchy-Euler Equations

  1. Reduction of Order for Linear Equations

    How reduction of order uses one known solution to find a second independent solution, where the reduction of order formula comes from, and why it is derived on an interval where the known solution is nonzero.

    Video 24

  2. Cauchy-Euler Equations

    How to solve Cauchy-Euler (equidimensional) equations with a power trial or a logarithmic change of variable, how repeated and complex exponents give logarithmic factors, and why the origin is a singular point.

    Video 25

  3. Order Reduction in Nonlinear Equations

    How the velocity substitution lowers the order of a nonlinear second-order equation with a missing dependent or independent variable, how first integrals arise, and where the position can serve as a local coordinate.

    Video 26