Chapter Outline

This chapter presents the matrix method for solving systems of first-order differential equations. These systems are instrumental in modeling applications with multiple interdependent processes, common in complex real-world situations.

6.1 Review of Matrices: This section offers a concise overview of essential matrix theory concepts in linear algebra, foundational for addressing systems of differential equations.

6.2 Review of Linear Independence and Systems of Equations: This section reviews the topic of systems of linear equations and methods for assessing the linear independence of solution sets.

6.3: Review: Eigenvalues and Eigenvectors: This section revisits eigenvalues and eigenvectors, explaining their calculation and importance in solving systems of differential equations.

6.4: Linear Systems of Differential Equations: This section introduces first-order differential equation systems and their matrix representations and discusses solution existence. It also explores transforming higher-degree differential equations into first-order system forms.

6.5 Solutions to Homogeneous Systems: This section details methods to find solutions for homogeneous differential equation systems and employs the Wronskian to verify solution independence.

6.6 Constant-Coefficient Homogeneous Systems: Real Eigenvalues: This section continues exploring homogeneous systems of differential equations with constant coefficients, focusing on scenarios with real-number eigenvalues.

6.7 Constant-Coefficient Homogeneous Systems: Complex Eigenvalues: This section addresses solutions for homogeneous systems with constant coefficients when eigenvalues are complex numbers.

6.8 Constant-Coefficient Homogeneous Systems: Repeated Eigenvalues: This section discusses solving homogeneous systems with constant coefficients when eigenvalues are repeated real numbers.

6.9 Nonhomogeneous Linear Systems: This section studies nonhomogeneous linear systems focusing on the method of variation of parameters.

Pioneers of Progress

Evelyn Boyd Granville, born in 1924 in Washington, D.C., is a pioneering mathematician whose journey is a testament to resilience and brilliance in the face of racial and gender barriers. As one of the first African-American women to earn a Ph.D. in mathematics from Yale University in 1949, Granville’s early work in functional analysis laid a foundation for her diverse and impactful career. She played a pivotal role in America’s space race, working with IBM on the Project Vanguard and Project Mercury space programs, where she developed complex computer algorithms for trajectory analysis. This work heavily relied on systems of differential equations to calculate the orbits and predict the paths of spacecraft – a critical component in the success of these early space missions.

Granville’s contributions extended beyond the realm of space exploration. She was also a passionate educator and advocate for women and minorities in STEM fields. Throughout her career, she taught mathematics at various universities and inspired countless students to pursue careers in science and technology.