Consider \(y''-5y'+6y=0\) and assume \(y(t)=e^{rt}\text{.}\) Solve this ODE in the usual way and also explore how it can be factored with differential operators. Notice the similarities in these two approaches.
Section 2.5 Method of Undetermined Coefficients
Subsection 2.5.1 Annihilators
Example 2.5.1.
Definition 2.5.2. Annihilators.
Let \(g\) be a suitably differentiable function of \(t\text{.}\) A differential operator \(L\) is an annihilator of \(g\) if \(L[g](t)=0\) for all \(t\text{.}\)
Example 2.5.3. Basic annihilators.
| Roots | \(g\) | Characteristic equation | Annihilator |
|---|---|---|---|
|
\(r=0\) mult \(n\) |
\(g(t)=c_0+c_1t+c_2t^2+\ldots+c_nt^{n-1}\) | \(r^n=0\) | \(D^n\) |
|
\(r=\alpha\) mult \(n\) |
\(g(t)=c_1e^{\alpha t}+c_2te^{\alpha t}+\ldots+c_nt^{n-1}e^{\alpha t}\) | \((r-\alpha)^n=0\) | \((D-\alpha)^n\) |
|
\(r=\alpha \pm \beta i\) mult \(n\) |
\(\begin{multline*}g(t)=e^{\alpha t}\left( a_0 \cos(\beta t)+b_0\sin(\beta t) \right) \\ + te^{\alpha t}\left(a_1\cos(\beta t)+b_1\sin(\beta t)\right)+ \ldots \\ + c_{n-1}t^{n-1}\left(a_{n-1}\cos(\beta t)+b_{n-1}\sin(\beta t)\right)\end{multline*}\) | \([(r-\alpha)^2+\beta^2]^n=0\) | \([(D-\alpha)^2+\beta^2]^n\) |
Algorithm 2.5.5. Finding smallest annihilators for a given \(g\).
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Identify the roots of \(g\text{.}\) Terms with the same roots go together.
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Build the smallest characteristic equation for \(g\text{.}\)
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Change the \(r\) into a \(D\) to get the smallest annihilator
Example 2.5.6.
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\(\displaystyle g(t)=3t\cos(2t)+8e^{2t}\)
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\(\displaystyle g(t)=4+7t^2+3e^{2t}\sin(t)\)
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\(\displaystyle g(t)=4e^{-t}\sin(2t)+5\cos(2t)\)
Subsection 2.5.2 Method of undetermined coefficients (annihilator approach)
Consider the linear nohomogeneous ODE with constant coefficients
\begin{equation}
ay''+by'+cy=g(t)\tag{2.5.1}
\end{equation}
For our first nonohomogeneous method, we want the terms of \(g(t)\) to contain functions associated with roots from the characteristic equation. Thus, finding a solution to (2.5.1) is equivalent to solving a higher-order linear homogeneous ODE with constant coefficients.
Definition 2.5.7. Homogeneous and nonohomogeneous solutions.
Solutions to (2.5.1) on \((-\infty,\infty)\) are of the form \(y=y_h+y_p\text{,}\) where \(y_h\) is the homogeneous solution, i.e. the solution that solves (2.5.1) when \(g(t)=0\) for all \(t\text{.}\) The homogeneous solution always contains one or more arbitrary constants. The term \(y_p\) is the nonhomogeneous solution, i.e. the solution that solves (2.3.1) when \(g(t)\neq 0\) for all \(t\text{.}\) The nonhomogeneous solution includes a known coefficient.
Algorithm 2.5.8. Method of undetermined coefficients.
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Find the homogeneous solution \(y_h\text{.}\)
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Identify the roots of \(g\text{.}\)
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Find the smallest characteristic equation for \(g\text{.}\)
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Apply the annihilator to both sides (on the left) to get a higher-order, linear, homogeneous ODE with constant coefficients.
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Solve by assuming \(y(t)=e^{rt}\) and write the roots right-to-left.
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Identify the terms of \(y_p\text{.}\)
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Substitute \(y_p\) into (2.5.1) to determine its coefficients.
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Write a general solution.
Example 2.5.9.
Consider the differential equation \(y''+4y=6t+2+5e^{2t}\text{.}\)
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Classify the differential equation by order, linearity, type of coefficients, and whether it is homogeneous or nonhomogeneous.
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What method(s) can you use to solve this differential equation?
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Find the general solution.
Example 2.5.10.
Consider the differential equation \(y'''-y=4e^t\text{.}\)
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Classify the differential equation by order, linearity, type of coefficients, and whether it is homogeneous or nonhomogeneous.
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What method(s) can you use to solve this differential equation?
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Find the general solution.
Subsection 2.5.3 Method of undetermined coefficients (superposition approach)
We don’t always have to use an annhilator with the method of undetermined coefficients. IF we compare the roots of \(y_h\) and \(y_p\text{,}\) we can identify any multiplicities and find the terms of \(y_p\text{.}\) We will determine the coefficients of \(y_p\) the same as before.
Example 2.5.11.
Consider the differential equation \(y''+4y=6\sin(2t)\text{.}\)
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Classify the differential equation by order, linearity, type of coefficients, and whether it is homogeneous or nonhomogeneous.
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What method(s) can you use to solve this differential equation?
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Find the general solution.
Example 2.5.12.
Consider the differential equation \(t^2y''-2ty'=4t\) with \(t>0\text{.}\)
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Classify the differential equation by order, linearity, type of coefficients, and whether it is homogeneous or nonhomogeneous.
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What method(s) can you use to solve this differential equation?
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Find the general solution.
