Characteristic Equations with Complex Roots

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

• Section 4.3 – Auxiliary Equations with Complex Roots

Expected Educational Results

• Objective 11–1: I can identify complex solutions to the characteristic equation for $n^{\text{th}}$$n^{\text{th}}$-degree homogeneous linear ODEs.

• Objective 11–2: I understand the form of the solution to an $n^{\text{th}}$$n^{\text{th}}$-degree homogeneous linear ODE for complex roots to the characteristic equation.

• Objective 11–3: I can find the most general solution to $n^{\text{th}}$$n^{\text{th}}$-degree homogeneous linear ODEs.

• Objective 11–4: I can find the solution to $n^{\text{th}}$$n^{\text{th}}$-degree homogeneous linear IVPs.

Homogeneous Equations

Constant Coefficient Method

Investigation 13

Solve the following $n^{\text{th}}$$n^{\text{th}}$-order homogeneous DEs using linear combination of $n$$n$ linearly-independent solutions:

1. ${\displaystyle y^{\prime \prime }-2y^{\prime }-2y=0}$$\displaystyle y^{\,\prime\prime}-2y^{\,\prime}-2y=0$

2. ${\displaystyle y^{\prime \prime }+3y^{\prime }+6y=0}$$\displaystyle y^{\,\prime\prime}+3y^{\,\prime}+6y=0$

3. ${\displaystyle 36y^{\prime \prime }-36y^{\prime }+13y=0}$$\displaystyle 36y^{\,\prime\prime}-36y^{\,\prime}+13y=0$

4. ${\displaystyle y^{\prime \prime }+y^{\prime }+y=0}$$\displaystyle y^{\,\prime\prime}+y^{\,\prime}+y=0$

5. ${\displaystyle y^{\prime \prime \prime }+3y^{\prime }=0}$$\displaystyle y^{\,\prime\prime\prime}+3y^{\,\prime}=0$

6. ${\displaystyle y^{(4)}+y^{\prime \prime }-2y=0}$$\displaystyle y^{\,(4)}+y^{\,\prime\prime}-2y=0$

7. ${\displaystyle y^{(4)}+32y^{\prime \prime }+256y=0}$$\displaystyle y^{\,(4)}+32y^{\,\prime\prime}+256y=0$

8. ${\displaystyle y^{(5)}-5y^{\prime \prime \prime }+4y^{\prime }=0}$$\displaystyle y^{\,(5)}-5y^{\,\prime\prime\prime}+4y^{\,\prime}=0$

9. ${\displaystyle y^{(5)}(t)+y^{(4)}(t)+4y^{\prime \prime \prime }(t)+3y^{\prime \prime }(t)+3y^{\prime }(t)=0}$$\displaystyle y^{\,(5)}(t)+y^{\,(4)}(t)+4y^{\,\prime\prime\prime}(t)+3y^{\,\prime\prime}(t)+3y^{\,\prime}(t)=0$

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Last Modified: Monday, 6 September 2020 13:33 EDT