Author: John J Weber III, PhDCorresponding 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 ${n}^{\text{th}}$-degree homogeneous linear ODEs.

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

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

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

Homogeneous Equations

Solving IVPs

Investigation 14

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

${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}+4{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}+5y=0$; $y(0)=0$ and ${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(0)=2$

${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}-2{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}+2y=0$; $y(0)=1$ and ${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(0)=0$