IntermediateTopic 69 of 104Engineering Mathematics
Systems of ODEs
Coupled equations written as x'=Ax and solved using eigenvalues.
In depth
A system x'=Ax is solved with eigenvalues: x(t)=Σcᵢvᵢe^{λᵢt}. The signs of real parts of λ determine stability; complex eigenvalues give oscillation. State-space models represent multi-variable systems, coupled tanks and multi-mass vibration.
Key points
- Phase portraits show trajectories
- Matrix exponential e^{At}
- Nonlinear systems linearised about equilibria
ExampleA predator-prey linearisation gives oscillating populations with complex eigenvalues.
Coupled systems, such as two masses linked by springs or interacting tanks, are written as matrix equations and solved with eigenvalues.
What is Systems of ODEs?
Coupled equations written as x'=Ax and solved using eigenvalues.
Why is it important to understand systems of ODEs?
Coupled systems, such as two masses linked by springs or interacting tanks, are written as matrix equations and solved with eigenvalues.
What is a state-space model?
It writes the system as x dot equals A x plus B u, with outputs y equals C x plus D u. It is the standard form for modern control and simulation.
Why are eigenvalues of A important?
They are the system poles, which determine natural frequencies, damping and stability.
How is Systems of ODEs used in practice?
Typical applications include modelling two tanks that exchange fluid through a pipe, analysing a two-degree-of-freedom vibration absorber and writing the state equations of a multi-state electrical network.
What should I learn before Systems of ODEs?
Study "Transfer functions" first (topic 68 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Systems of ODEs?
Phase portraits show trajectories. Matrix exponential e^{At}. Nonlinear systems linearised about equilibria.
Can you give an example of Systems of ODEs?
A predator-prey linearisation gives oscillating populations with complex eigenvalues.
What comes after Systems of ODEs?
The next topic is "Fourier series", which builds directly on this one.
How difficult is Systems of ODEs?
It is classed as Intermediate in Engineering Mathematics (topic 69 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.