Undetermined coefficients is a method to solve non-homogeneous linear differential equations.
The solution of a nonhomogeneous equation can be expressed as the sum of the solution to its corresponding homogeneous equation and a particular solution. For example, consider the following equation:
Let
We can see why this is true by doing some algebraic manipulation:
Examples
Example: Solve
First we solve the homogeneous equation
\begin{align} r^2 - 3r - 4 &= 0 \ (r - 4)(r + 1) &= 0 \end{align}
\begin{align} 4Ae^{2x} - 6Ae^{2x} - 4Ae^{2x} &= 3e^{2x} \ A &= -\frac{1}{2} \ y_p &= -\frac{1}{2}e^{2x} \end{align}
y = C_1 e^{4x} + C_2 e^{-x} - \frac{1}{2}e^{2x}
Example: solve
From the above example,
Let , then
\begin{align} y_p’ &= 2Ax + B \ y_p” &= 2A \ \end{align}
\begin{align} 2A - 3(2Ax + B) - 4(Ax^2 + Bx + C) &= 4x^2 \ -4Ax^2 + (-6A - 4B)x + 2A - 3B - 4C &= 4x^2 \ A = -1, B = \frac{3}{2}, C = -\frac{13}{8} \end{align}
y = C_1 e^{4x} + C_2 e^{-x} -x^2 + \frac{3}{2}x - \frac{13}{8}