# 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.

## Complex Solutions to Characteristic Equations

Let $z\left(t\right)=u\left(t\right)+iv\left(t\right)$$z(t)=u(t)+iv(t)$ be a solution to $a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+cy=0$$\displaystyle ay^{\,\prime\prime}+by^{\,\prime}+cy=0$, where $a,b,c\in \mathbb{R}$$a, b, c\in\mathbb{R}$. Then the real part $u\left(t\right)$$u(t)$ and the imaginary part $v\left(t\right)$$v(t)$ are real-valued solutions.

If $r=\alpha ±i\beta$$r=\alpha\pm i\beta$ are complex conjugate solutions to $a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+cy=0$$\displaystyle ay^{\,\prime\prime}+by^{\,\prime}+cy=0$, then two linearly independent solutions are ${e}^{\alpha t}\mathrm{cos}\left(\beta t\right)$$e^{\alpha t}\cos{(\beta t)}$ and ${e}^{\alpha t}\mathrm{sin}\left(\beta t\right)$$e^{\alpha t}\sin{(\beta t)}$, and the general solution is\newline $y\left(t\right)={c}_{1}{e}^{\alpha t}\mathrm{cos}\left(\beta t\right)+{c}_{2}{e}^{\alpha t}\mathrm{sin}\left(\beta t\right)$$y(t)=c_1e^{\alpha t}\cos{(\beta t)}+c_2e^{\alpha t}\sin{(\beta t)}$, where ${c}_{1}$$c_1$ and ${c}_{2}$$c_2$ are arbitrary constants.

#### Investigation 09

Let $y={e}^{rt}$$y=e^{rt}$, where $r=\alpha ±i\beta$$r=\alpha\pm i\beta$, be solutions to $a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+cy=0$$ay^{\,\prime\prime}+by^{\,\prime}+cy=0$.

1. Use Euler's formula to rewrite ${y}_{a}={e}^{\left(\alpha +i\beta \right)t}$$y_a=e^{(\alpha + i\beta)t}$.

2. Use Euler's formula to rewrite ${y}_{b}={e}^{\left(\alpha -i\beta \right)t}$$y_b=e^{(\alpha - i\beta)t}$.

3. Evaluate ${y}_{1}=\frac{1}{2}{y}_{a}+\frac{1}{2}{y}_{b}$$y_1=\frac{1}{2}y_a+\frac{1}{2}y_b$. Explain why $\frac{1}{2}$$\frac{1}{2}$ is a valid coefficient for ${y}_{a}$$y_a$ and ${y}_{b}$$y_b$.

4. Evaluate ${y}_{2}=\frac{1}{2i}{y}_{a}-\frac{1}{2i}{y}_{b}$$y_2=\frac{1}{2i}y_a-\frac{1}{2i}y_b$. Explain why $\frac{1}{2i}$$\frac{1}{2i}$ is a valid coefficient for ${y}_{a}$$y_a$ and ${y}_{b}$$y_b$.

5. Explain why ${y}_{1}$$y_1$ and ${y}_{2}$$y_2$ are solutions to $a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+cy=0$$ay^{\,\prime\prime}+by^{\,\prime}+cy=0$.

6. Explain why $y={c}_{1}{y}_{1}+{c}_{2}{y}_{2}$$y=c_1y_1+c_2y_2$ is a solution to $a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+cy=0$$ay^{\,\prime\prime}+by^{\,\prime}+cy=0$.

#### Investigation 10

Suppose the following are roots to the characteristic equation. Find the homogeneous solution, ${y}_{h}\left(x\right)$$y_h(x)$.

1. $r=-2i,2i$$\displaystyle r=-2i,2i$

2. $r=0,-3i,3i$$\displaystyle r=0,-3i,3i$

3. $r=1,i\sqrt{3},-i\sqrt{3}$$\displaystyle r=1,i\sqrt{3},-i\sqrt{3}$

4. $r=-3i,-3i,3i,3i$$\displaystyle r=-3i,-3i,3i,3i$

5. $r=1,1,-2i,2i$$\displaystyle r=1,1,-2i,2i$

6. $r=1-2i,1+2i$$\displaystyle r=1-2i,1+2i$

7. $r=2,2,2-4i,2+4i$$\displaystyle r=2,2,2-4i,2+4i$

8. $r=\frac{3-2i}{2},\frac{3+2i}{2}$$\displaystyle r=\frac{3-2i}{2},\frac{3+2i}{2}$

#### Investigation 11

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$

#### Investigation 12

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

1. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+4{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+5y=0$$\displaystyle y^{\,\prime\prime}+4y^{\,\prime}+5y=0$; $y\left(0\right)=0$$y(0)=0$ and ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=2$$y^{\,\prime}(0) = 2$

2. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }-2{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+2y=0$$\displaystyle y^{\,\prime\prime}-2y^{\,\prime}+2y=0$; $y\left(0\right)=1$$y(0)=1$ and ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=0$$y^{\,\prime}(0) = 0$

3. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }\left(t\right)+3{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+6{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)+8y\left(t\right)=0$$\displaystyle y^{\,\prime\prime\prime}(t)+3y^{\,\prime\prime}(t)+6y^{\,\prime}(t)+8y(t)=0$, $y\left(0\right)=1$$y(0)=1$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=\sqrt{3}$$y^{\,\prime}(0)=\sqrt{3}$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(0\right)=-2$$y^{\,\prime\prime}(0)=-2$