NONHOMOGENEOUS EQUATIONS;
UNDERTERMINED COEFFICIENTS METHOD

Consider equations of the form
$L[y]=\frac{d^2y}{dt^2}+ p(t)\frac{dy}{dt}+ q(t)y = g(t),$
given independent solutions y₁(t) and y₂(t) to L[y]=0.

General Solution

is y(t) = c₁y₁(t)+c₂y₂(t)+Y(t),
where Y(t) is a particular solution to L[y]=g(t).

Superposition of Solutions:

if $g = g_1 + \cdots + g_n$,
and L[Y_i] = g_i, for $i=1,\ldots ,n$, then $Y = Y_1 + \cdots + Y_n$.

Undetermined Coefficients:

for $L[y] = a\frac{d^2y}{dt^2}+ b\frac{dy}{dt}+ cy$.

• Consider each g_i separately and sum results to get Y.
• Standard cases for g_i(t):
  • If $g_i(t) = P_n(t) = p_0t^n + \cdots + p_n$, then
    $Y_i(t) = t^sQ_n(t)= t^s(q_0t^n + \cdots + q_n)$, $s = 0,1\mbox{\ or\ }2$.
  • If $g_i(t) = P_n(t)e^{\alpha t}$, then $Y_i(t) = t^sQ_n(t)e^{\alpha t}$.
  • If $g_i(t)=P_n(t)e^{\alpha t}(\sin(\beta t)\mbox{\ or\ }\cos(\beta t))$,
    then $Y_i(t) = t^se^{\alpha t}(Q_n(t)\sin(\beta t)+R_n(t)\cos(\beta t))$.

• Steps for complete solution:

  1.

  Find y₁(t) and y₂(t), solutions to L[y]=0.

  2.

  Split g(t) into terms g_i(t).

  3.

  For each g_i(t), find Y_i(t), and add Y_i(t)'s to get Y(t).

  4.

  Apply initial conditions to determine c₁ and c₂, and
  final solution y(t) = c₁y₁(t)+c₂y₂(t)+Y(t).