Differential Operators and the Elimination Method for Systems

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 5.2 – Differential Operators and the Elimination Method for Systems

Expected Educational Results

• Objective 15–1: I can identify the linear differential operator for any polynomial function, exponential function, or sine and cosine functions.
• Objective 15–2: I can solve linear systems of ODEs using linear differential operators.

Differential Operators

The Annihilator Method for Solving DEs

Steps for the Annihilator Method for Solving DEs. For example, solve $ay^{\,\prime\prime}+by^{\,\prime}+cy=f(x)$

1. Write $ay^{\,\prime\prime}+by^{\,\prime}+cy$ in terms of $L(D)$.
2. Find $A(D)$ for $f(x)$
3. Rewrite $ay^{\,\prime\prime}+by^{\,\prime}+cy=f(x)$ as $A(D)L(D)[y]=A(D)f(x)=0$. [Explain why $A(D)f(x)=0$].
4. Find the solutions to $A(D)L(D)[y]$. [You will need to know which $c_i$ are coefficients of terms in $y_p(x)$. Why?]
5. Write the general solution to the DE.

Investigation 12

Use linear differential operators and annihilators to solve the following DEs:

NOTE: In each ode below, the Method of Undetermined Coefficients and Variation of Parameters may be easier to use. This question is to recognize that using linear differential operators and annihilators will result in the same solutions.

1. $\displaystyle y^{\,\prime\prime}+6y^{\,\prime}+y=e^{3x}$
2. $\displaystyle y^{\,\prime\prime}+6y^{\,\prime}+y=\sin{(x)}$
3. $\displaystyle y^{\,\prime\prime}+6y^{\,\prime}+y=e^{3x}-\sin{(x)}$
4. $\displaystyle y^{\,\prime\prime\prime}+9y^{\,\prime}=x^2\cos{(3x)}$
5. $\displaystyle y^{\,(5)}-y^{\,\prime\prime\prime}=e^x+3x^2-2$
6. $\displaystyle y^{\,(4)}-2y^{\,\prime\prime}+y=x^2\sin(x)$
7. $\displaystyle y^{\,\prime\prime}-6y^{\,\prime}+13y=xe^{3x}\cos{(2x)}$

Example 04: Solve: $\displaystyle y^{\,\prime\prime}+5y^{\,\prime}+6y=e^{3x}$

Solution:

Rewrite left-side of ode using linear differential operator, $L(D)$:

$\displaystyle (D^2+5D+6)[y(x)] = e^{3x}$

Find the annihilator, $A(D)$, for the nonhomogeneous function, $e^{3x}$, and apply to both sides of the ode:

$\displaystyle (D-3)(D^2+5D+6)[y(x)] = (D-3)[e^{3x}]$

The above equation is $\displaystyle A(D)L(D)[y(x)] = 0$

Find the homogeneous solution $y_h(x)$:

The characteristic equation is $r^2+5r+6=0$ and has roots $r=-2,-3$. So, $\displaystyle y_h(x)=c_1e^{-2x} + c_2e^{-3x}$.

Find the particular solution $y_p(x)$:

From the annihilator, $A(D)$, $\displaystyle y_p(x)=Ae^{3x}$. You must always use algebra to find the coefficients of $y_p$, in this example $\displaystyle A=\frac{1}{30}$.

Thus, the solution to the ode is: $\displaystyle y(x)=y_h(x)+y_p(x)=c_1e^{-2x} + c_2e^{-3x}+\frac{1}{30}e^{3x}$.

Investigation 13

What do you notice about $L(D)$ and the characteristic equation in Example 04? Explain.

Investigation 14

What do you notice about $A(D)$ and the particular solution? Explain.

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Last Modified: Monday, 19 October 2020 8:23 EDT