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. ${\displaystyle y^{\prime \prime }+4y^{\prime }+5y=0}$$\displaystyle y^{\,\prime\prime}+4y^{\,\prime}+5y=0$; $y(0)=0$$y(0)=0$ and $y^{\prime }(0)=2$$y^{\,\prime}(0) = 2$

2. ${\displaystyle y^{\prime \prime }-2y^{\prime }+2y=0}$$\displaystyle y^{\,\prime\prime}-2y^{\,\prime}+2y=0$; $y(0)=1$$y(0)=1$ and $y^{\prime }(0)=0$$y^{\,\prime}(0) = 0$

3. ${\displaystyle y^{\prime \prime \prime }(t)+3y^{\prime \prime }(t)+6y^{\prime }(t)+8y(t)=0}$$\displaystyle y^{\,\prime\prime\prime}(t)+3y^{\,\prime\prime}(t)+6y^{\,\prime}(t)+8y(t)=0$, $y(0)=1$$y(0)=1$, $y^{\prime }(0)=\sqrt{3}$$y^{\,\prime}(0)=\sqrt{3}$, $y^{\prime \prime }(0)=-2$$y^{\,\prime\prime}(0)=-2$

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