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. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }-2{y}^{\phantom{\rule{0.167em}{0ex}}\prime }-2y=0$$\displaystyle y^{\,\prime\prime}-2y^{\,\prime}-2y=0$

2. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+3{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+6y=0$$\displaystyle y^{\,\prime\prime}+3y^{\,\prime}+6y=0$

3. $36{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }-36{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+13y=0$$\displaystyle 36y^{\,\prime\prime}-36y^{\,\prime}+13y=0$

4. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+y=0$$\displaystyle y^{\,\prime\prime}+y^{\,\prime}+y=0$

5. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }+3{y}^{\phantom{\rule{0.167em}{0ex}}\prime }=0$$\displaystyle y^{\,\prime\prime\prime}+3y^{\,\prime}=0$

6. ${y}^{\phantom{\rule{0.167em}{0ex}}\left(4\right)}+{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }-2y=0$$\displaystyle y^{\,(4)}+y^{\,\prime\prime}-2y=0$

7. ${y}^{\phantom{\rule{0.167em}{0ex}}\left(4\right)}+32{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+256y=0$$\displaystyle y^{\,(4)}+32y^{\,\prime\prime}+256y=0$

8. ${y}^{\phantom{\rule{0.167em}{0ex}}\left(5\right)}-5{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }+4{y}^{\phantom{\rule{0.167em}{0ex}}\prime }=0$$\displaystyle y^{\,(5)}-5y^{\,\prime\prime\prime}+4y^{\,\prime}=0$

9. ${y}^{\phantom{\rule{0.167em}{0ex}}\left(5\right)}\left(t\right)+{y}^{\phantom{\rule{0.167em}{0ex}}\left(4\right)}\left(t\right)+4{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }\left(t\right)+3{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+3{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)=0$$\displaystyle y^{\,(5)}(t)+y^{\,(4)}(t)+4y^{\,\prime\prime\prime}(t)+3y^{\,\prime\prime}(t)+3y^{\,\prime}(t)=0$