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3: Systems of ODEs

Last updated

Jun 16, 2022
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- 2.E: Higher order linear ODEs (Exercises)
- 3.1: Introduction to Systems of ODEs

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3.1: Introduction to Systems of ODEs
Often we do not have just one dependent variable and just one differential equation, we may end up with systems of several equations and several dependent variables even if we start with a single equation.
3.2: Matrices and linear systems
This page provides a comprehensive review of matrices and vectors, essential for understanding linear systems of ordinary differential equations (ODEs). It covers various matrix operations, including multiplication by a scalar, matrix addition, and matrix multiplication, explaining the dot product and identity matrix. The section also introduces determinants, providing methods for calculation and the significance, namely its role in determining matrix invertibility.
3.3: Linear systems of ODEs
This page provides an overview of matrix and vector valued functions, particularly focusing on their differentiation. It explains how matrix valued functions are structured and discusses their differentiation rules. The page further delves into systems of linear ordinary differential equations (ODEs), specifically first-order systems, their solutions, and the principle of superposition for homogeneous systems.
3.4: Eigenvalue Method
In this section we will learn how to solve linear homogeneous constant coefficient systems of ODEs by the eigenvalue method.
3.5: Two dimensional systems and their vector fields
The page provides an overview of constant coefficient linear homogeneous systems in the plane, focusing on eigenvalues and eigenvectors, and classifying the system behavior based on the eigenvalues. It explains scenarios with real positive, real negative, and opposite sign eigenvalues, resulting in sources, sinks, or saddles, respectively. It also covers complex eigenvalues, detailing their effect on system behavior, such as producing centers or spirals (source or sink).
3.6: Second order systems and applications
The page discusses undamped mass-spring systems, focusing on the equations of motion for interconnected masses via springs and the resulting models that arise. It begins by outlining the dynamics of a three-mass system using Hooke???s and Newton???s laws, providing matrix representations for mass and stiffness. It then explores solutions using eigenvectors and eigenvalues, expanding to more complex systems or additional forces.
3.7: Multiple Eigenvalues
Often a matrix has ???repeated??? eigenvalues. That is, the characteristic equation det(A?????I)=0 may have repeated roots. As any system we will want to solve in practice is an approximation to reality anyway, it is not indispensable to know how to solve these corner cases. It may happen on occasion that it is easier or desirable to solve such a system directly.
3.8: Matrix exponentials
In this section we present a different way of finding the fundamental matrix solution of a system.
3.9: Nonhomogeneous systems
The document explores various methods for solving nonhomogeneous first order systems with special focus on matrices and vectors. It covers the integrating factor method to derive solutions and solve examples, demonstrating it with matrix transformations and eigenvectors. The text also delves into the method of variation of parameters, which is useful when coefficient matrices aren't constant.
3.E: Systems of ODEs (Exercises)
These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.

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