Undetermined Coefficients

For non-homogeneous linear equations with constants coefficients:

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=f(t),\text{ where }f(t)\ne 0$$

A particular solution ($y_p$) is a function that contains no arbitrary constants and satisfies that equation

• this is different than the particular solution meaning the solution to an IVP for first order DEQs

Method

1. Use the form of $f$ to guess a form for a particular solution
2. Plug the guessed form of the particular solution into the DE
3. Determine the coefficients that make your guessed solution an actual equation

Constraints

• The DEQ is Linear
• Coefficients are constants
• $f(t)$ is one of:
  • Constants
  • Polynomials
  • Exponential function
  • Sine or Cosine
  • A combination of the above (through multiplication and addition)
• Not limited to second order DEs

Guessing Form

$f(t)$	$y_p(t)$
$P(t)$	$t^{s}Q(t),\text{ deg }Q=\text{ deg }P$
$e^{\alpha t}$	$A{t}^s{e}^{\alpha t}$
$e^{\alpha t}P(t)$	$t^s{e}^{\alpha t}Q(t),\text{ deg }Q=\text{ deg }P$
$e^{\alpha t}(\cos /\sin )(\beta t)$	$t^s{e}^{\alpha t}(A\cos (\beta t)+B\sin (\beta t))$
$P(t{)}_k{e}^{\alpha t}(\cos /\sin )(\beta t)$	$t^s{e}^{\alpha t}(A_k(t)\cos (\beta t)+B_k(t)\sin (\beta t))$

• If a term of the form overlaps with a term of the Homogeneous Linear Equation solution, multiply that term by $t$ by setting $s>0$
• When there are multiple polynomials, use the term of the highest one as the term of all