11 Undetermined Coefficients
The undetermined coefficient method is possible for a few standards functions such as \(R(x) = e^{ax}, \sin(ax), \cos(ax),\) polynomials. This method also requires a trial solution to compute the required particular solution.
Note If RHS \(= \sin(2x) + \cos(2x)\), then we can consider it as a single function \(R(x)\)
Consider
- Constants \(l, k, a, b \in R\)
- Undetermined Coefficients \(A, B, A_0, A_1, \dots, A_n\) (unknown constant)
The exception cases are for preventing duplication of terms, and hence prevent linear dependency of the solutions.
| \(R(x)\) | Trial Particular Solution | Exception based on root \(m\) of auxilary eqn |
|---|---|---|
| \(l e^{ax}\) | \(A e^{ax}\) | |
| \(A x e^{ax}\) | \(m_1 = a\) or \(m_2 = a\) | |
| \(A x^2 e^{ax}\) | \(m_1 = m_2 = a\) | |
| \(l \cos(ax)\) \(l \sin(ax)\) \(l \cos(ax) \pm k \sin(ax)\) | \(A \cos ax + B \sin ax\) | |
| \(x (A \cos ax + B \sin ax)\) | \(m= 0 \pm ai\) | |
| \(a_0 + a_1 x + \dots + a_n x^n\) (\(n^{\text{th}}\) degree polynomial) | \((A_0 + A_1 x + \dots + A_n x^n)\) | |
| \(x(A_0 + A_1 x + \dots + A_n x^n)\) | \(m = 0\) | |
| \(e^{ax} \cos bx\) \(e^{ax} \sin bx\) \(e^{ax} ( \cos bx + \sin bx )\) | \(e^{ax} ( A \cos bx + B \sin bx )\) | |
| \(xe^{ax} ( A \cos bx + B \sin bx )\) | \(m = a \pm bi\) |
Trick for product of 3 functions¶
If \(y_g\) and the trial particular solution are similar
- instead of using \((uvw)' = uvw' + uv'w + u'vw\)
- we can take
\[ x e^x ( A \cos x + B \sin x) \to x \phi \]
Example¶
\[ \begin{aligned} y'' - 2y' + 2y &= e^x \sin x \\ x e^x ( A \cos x + B \sin x) & \to x \phi \\ {y_g}'' - 2{y_g}' + 2{y_g} &= 0 \\ \implies \phi'' - 2\phi' + 2\phi &= 0 \end{aligned} \]