DE-BOOK Key Terms & Concepts

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Worksheet Key Terms & Concepts

✳️ Summary of the Key Ideas.

Linear Homogeneous Differential Equations with Constant Coefficients (LHCC)
- These are differential equations where each term consists of a derivative of the unknown function multiplied by a constant.
  
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- The general form of an LHCC equation is:
  \begin{equation*} a_n\ y^{(n)} + a_{n-1}\ y^{(n-1)} + \dots + a_1\ y' + a_0\ y = 0\text{.} \end{equation*}
  
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The Characteristic Equation
- By assuming a solution of the form \(y = e^{rx}\text{,}\) an LHCC can be reduced to a characteristic polynomial in \(r\text{.}\)
  
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- The solutions to the characteristic equation determine the form of the general solution.
  
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Solution Types
- Let \(r\) be a solution to the characteristic equation (CE).
  
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- If \(r\) is different from all other solutions of the CE, then
  \begin{equation*} c e^{r x} \end{equation*}
  is a term of the general solution.
  
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- If \(r\) is equal to, say, three other solutions of the CE, then
  \begin{equation*} c_1 e^{r x} + c_2 x e^{r x} + c_3 x^2 e^{r x} \end{equation*}
  are terms of the general solution.
  
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- If \(r = \alpha + i\beta\) or \(r = \alpha - i\beta\text{,}\) then the general solution contains
  \begin{equation*} e^{\alpha x}(c_1\sin(\beta x)+c_2\cos(\beta x))\text{.} \end{equation*}
  
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✍🏻 Method 7. LHCC Method.

The general solution to a linear homogeneous differential equation with constant coefficients (LHCC) of the form

\begin{equation} a_n\ y^{(n)} + a_{n-1}\ y^{(n-1)} + \cdots + a_2\ y'' + a_1\ y' + a_0\ y = 0,\tag{51} \end{equation}

can be found through the following steps...

Step 1: Solve the Characteristic Equation 🔗

Solve the characteristic equation (CE)

\begin{equation*} a_n\ r^{n} + a_{n-1}\ r^{n-1} + \cdots + a_2\ r^2 + a_1\ r + a_0 = 0, \end{equation*}

Step 2: Write Down the General Solution 🔗

Real & Different: \(r_1, r_2, \dots, r_n \)
\begin{equation*} y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x} + \dots + c_n e^{r_n x}\text{.} \end{equation*}

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Real & Repeated: \(r_1 \) (multiplicity \(m \))
\begin{equation*} y(x) = (c_1 + c_2 x + \dots + c_m x^{m-1}) e^{r_1 x}\text{.} \end{equation*}

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Complex: \(\alpha \pm i\beta \)
\begin{equation*} y(x) = e^{\alpha x} \left(c_1 \cos(\beta x) + c_2 \sin(\beta x)\right)\text{.} \end{equation*}

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For mixed root types, combine the corresponding terms to form the complete general solution.

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