Type a command or search the notes. Tab cycles completions. Up and down recall history when blank. Escape dismisses, then leaves the prompt.
MATH
[1251]

Theorems from Second-order constant-coefficient equations

All theorems from Second-order constant-coefficient equations.

Back to Second-order constant-coefficient equations

Superposition of homogeneous solutions

For a homogeneous linear equation, 00 is a solution, scalar multiples of solutions are solutions and sums of solutions are solutions. For a second-order equation, two linearly independent solutions generate every homogeneous solution:

yh=Ay1+By2.y_h=Ay_1+By_2.

The two arbitrary constants are what allow two initial conditions to be imposed. If the proposed solutions are scalar multiples, they provide only one mode and cannot represent all possible initial data.

  1. [1]:Constant-coefficient equation

Homogeneous part plus particular part

Write a linear equation as

L[y]=f(x),L[y]=ay′′+by′+cy.L[y]=f(x), \qquad L[y]=ay''+by'+cy.

If ypy_p is one particular solution and yhy_h is any homogeneous solution, then the complete solution is

y=yh+yp.y=y_h+y_p.

The homogeneous part contains the arbitrary constants. The particular part accounts for the forcing and contains no new arbitrary constants.

  1. [1]:Constant-coefficient equation
  2. [2]:Proof
  3. [3]:Undetermined coefficients
  4. [4]:Periodic forcing

On this page

Superposition of homogeneous solutionsHomogeneous part plus particular part