082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

20 Ordinarydifferential
equationsII
Contents 20.1 Introduction 603
20.2 Analoguesimulation 603
20.3 Higherorderequations 606
20.4 State-spacemodels 609
20.5 Numericalmethods 615
20.6 Euler’smethod 616
20.7 ImprovedEulermethod 620
20.8 Runge--Kuttamethodoforder4 623
Reviewexercises20 626
20.1 INTRODUCTION
Engineerssometimesfinditconvenienttorepresentadifferentialequationusinganelectronic
circuit. This is known as analogue simulation. Analogue controllers are also frequentlyusedincontrolsystemsandthesearebasedonsimilarcircuits.Complicatedengineering
systems need to be modelled using a set of differential equations. This leads
tothetopicofstate-spacemodelling.Suchmodelsfindwidespreaduseintheaerospace
industries as well as in other areas that have complex systems. Numerical techniques
are very important as they allow differential equations to be solved using computers.
Often this is the only way to obtain a solution of a differential equation, owing to its
complexity. State-space models arealmost always solved using computers.
20.2 ANALOGUESIMULATION
Itispossibletosolvedifferentialequationsusingelectroniccircuitsbasedonoperational
amplifiers. The advantage of this approach is the ease with which the coefficients of
the differential equation can be adjusted and the effect on the solution observed. The