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Worksheet Key Terms & Concepts
β³οΈ Summary of the Key Ideas.
The method of undetermined coefficients is used to solve non-homogeneous linear differential equations.
The general solution to a non-homogeneous equation is the sum of the general solution to the corresponding homogeneous equation and a particular solution.
The method involves guessing the form of the particular solution based on the form of the non-homogeneous term and solving for the coefficients.
The method is applicable when the non-homogeneous term can be expressed as a linear combination of known functions.
LHCC Equation
An LHCC equation is a Linear Homogeneous Differential Equations with Constant Coefficients and have the form
\begin{equation*}
a_n y^{(n)} + \cdots + a_1 y' + a_0 y = 0\text{.}
\end{equation*}
Characteristic Equation
LHCC General Solutions
LNCC Equation
An LNCC equation is a Linear Nonhomogeneous Differential Equations with Constant Coefficients and have the form
\begin{equation}
a_n y^{(n)} + \cdots + a_1 y' + a_0 y = f(x)\text{.}\tag{54}
\end{equation}
Forcing Function
A forcing function is the free term in
(54) , denoted by
\(f(x)\text{.}\)
Particular Solution
A particular solution,
\(y_h\text{,}\) is a function that can be plugged into
(54) and all the terms combine to give the forcing function,
\(f(x)\text{.}\)
Homogeneous Solution
A homogeneous solution,
\(y_h\text{,}\) is a function such that, when it is plugged into
(54) all the terms cancel to zero. That is,
\(y_h\) is the solution to the homogeneous version of
(54) (
\(f(x) = 0 \) ).
General Solutions
The general solution to
(54) is the sum of the homogeneous and particular solutions, that is
\begin{equation*}
y = y_h + y_p\text{.}
\end{equation*}
\(y\) \(y'' - 4y' + 3{\color{blue} y} =\!\!\!\) LHS
LHS \(\os{?}{=} 9x\)
1
\(3\) \((3)'' - 4(3)' + 3({\color{blue} 3}) =\!\!\!\) \({\color{blue} 9}\) No
2
\(3x\) \((3x)'' - 4(3x)' + 3({\color{blue} 3x}) =\!\!\!\) \(-12 + {\color{blue} 9x}\) No
3
\(x^4\) \((x^4)'' - 4(x^4)' + 3({\color{blue} x^4}) =\!\!\!\) \(12x^2 - 16x^3 + {\color{blue} 3x^4}\) No
4
\(x^2+3x\) \((x^2+3x)'' - 4(x^2+3x)' + 3({\color{blue} x^2}+3x) =\!\!\!\) \(-10 + x + {\color{blue} 3x^2}\) No
5
\(3x-6\) \((3x-6)'' - 4(3x-6)' + 3({\color{blue} 3x}-6) =\!\!\!\) \(-30+{\color{blue} 9x}\) No
6
\(3x+4\) \((3x+4)'' - 4(3x+4)' + 3({\color{blue} 3x}+4) =\!\!\!\) \({\color{blue} 9x}\) Yes
Based on this table, we note that the correct solution is \(y = 3x + 4\) and
Row 1 shows that a free term alone could never produce an
\(x\) term.
Derivatives reduce the power of a polynomial, so the highest power term (highlighted in blue) comes from the
\(y\) term.
Rows 3 & 4 illustrate the solution canβt have a
\(x^2\) or higher-degree term.
Row 6 shows the solution (
\(3x+4\) ) needed an
\(x\) term and free term even though the right-hand side,
\(9x\text{,}\) has only an
\(x\) term.
Subsection LNCC Equations
Figure 328. Comparison of a homogeneous equation (left ) and a nonhomogeneous equation (right ). In both, the solutions must simplify in a specific way when substituted into the equation. Table 329. row 1
\(f(x)\ \) type
\(y_p\) Form
1
\(a\) (constant)
\(A\)
Table 330. row 2
\(f(x)\ \) type
\(y_p\) Form
2
\(ax + b\) \(Ax + B\)
Table 331. row 3
\(f(x)\ \) type
\(y_p\) Form
3
\(ax^2 + bx + c\) \(Ax^2 + Bx + C\)
Table 332. row 4
\(f(x)\ \) type
\(y_p\) Form
4
\(ax^3 + bx^2 + cx + d\) \(Ax^3 + Bx^2 + Cx + D\)
Table 333. row 5
\(f(x)\ \) type
\(y_p\) Form
5
\(a e^{\ds\alpha x}\) \(A e^{\ds\alpha x}\)
Table 334. row 6
\(f(x)\ \) type
\(y_p\) Form
6
\(a \sin(\beta x) + b \cos(\beta x)\) \(A \sin(\beta x) + B \cos(\beta x)\)
