# 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

### Solving IVPs

#### Investigation 14

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$