Differential Equations notes
16 notes by 9 users
This Differential Equations notes page gathers user-made notes on ordinary equations, linearisation and the logistic model. It is for university students practising methods and teachers looking for ready-made material. There are over 10 notes, mostly at university level. Choose a note to begin.
Notes
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Chapter 1 - Second Order Linear Differential Equations
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Ordinary Differential Equations Q1 - ODE's Introduction The relationship between independent variable 'x' and dependent variable 'y'. The derivative, dy/dx is the rate of change of something (first order differential equation) The second derivative,...
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Differential Equations - created from Mind Map_1 Classification Ordinary (ODE) or Partial (PDE) Order - order of the highest derivative of the dependent variable with respect to the independent variable Linear - only if the unknown function and its derivatives appea...
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Harvesting We can have constant harvesting, where a fixed amount is harvested, regardless of population size, and harvesting where a fixed fraction of the population is removed.Constant Harvesting with the Exponential Model - Th...
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Chapter 2 - Power Series Solution 2.1 Review of Power and Taylor SeriesTaylor Series are an example of a power series and is a generalisation of MacLaurin series. In order to talk about power series solutions to ODEs we need to series to converge. We ...
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Qualitative Properties of Solutions of Autonomous ODEs An autonomous ODE is simply an ODE in which the independent variable does not appear explicitly ie the "t" only appears on the d/dt side. dy/dt = f(y)The value y(t) can be represented as a point on the real number lin...
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The Logistic Model The exponential model only works when the population has unlimited resources. If there is a finite amount of resources then there will be a maximum population size that can be supported. This maximum is called the car...
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Untitled Q2 - Laplace Transforms Introduction The laplace transform method maps out an equation from it's original domain onto a new domain, making it easier to solve. Once solved, the equation can be transformed back to it's ...
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Method of Undetermined Coefficients Complementary Function1) Create auxiliary equation for homogeneous equation (=0)2) Solve for roots3) Based on roots form CFParticular Integral (skip step if DE was already homogeneous)4) Use data book to find trial PI...
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0. Introduce a change of variable to make the unknown homogeneous - created from Mind Map 1. Determine the appropriate orthogonal basis 2. Write the unknown as a linear combination of eigen function and substitute it in the differential equation 3. Expand the rhs in terms of orthogonal basis and equate coe...
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Linearisation Linearisation is the process of trying to linearise a nonlinear function. This is so that we can understand what is happening at equilibrium points. First, find equilibrium points where f(Y) = 0.Next, differentiate th...
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Differential Equations - created from Mind Map Classification Ordinary (ODE) or Partial (PDE) Order - order of the highest derivative of the dependent variable with respect to the independent variable Linear - only if the unknown function and its derivatives appea...
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UNDETERMINED COEFFICIENT - created from Mind Map Only applicable when and only when the right member of the equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. we need to look at g(t) and make a guess...
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Solving Ordinary Differential Equations - created from Mind Map Seperable Equations Linear First Order ODES solved with method don't know what it's called Solving Higher order Equations Solve Characteristic Equation and get homogenous solution Find Particular Solution
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Linear ODE's defined Linear ODE'sLinear ordinary differential equations have dependant and independent variables just like any equation. The independent variable is time (t)ALL linear ODE's must have only linear combination...
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Ordinary Differential Equations - created from Mind Map is single variable DE - depens on only x Separable DE Linear ODE
More Differential Equations resources
About differential equations notes
GoConqr OriginalExplore Differential Equations notes
These notes cover definitions, methods and models. Ordinary Differential Equations introduces the relationship between independent and dependent variables and the derivative. The notes on second order linear equations begin with a general form, while Method of Undetermined Coefficients sets out steps for finding a solution.
For modelling, The Logistic Model discusses a limit on population size. Linearisation looks at understanding equilibrium points, and Harvesting describes different ways a population can be reduced.
How to use the notes
Start with Ordinary Differential Equations if you want to revisit the language of derivatives before choosing a method. Read a section, then write down its main idea in your own words. When you need to check recall or see connections between ideas, switch to a quiz or mind map on the unit page. Pick a note and work through a section at your own pace.
Frequently asked questions
Which note is a useful starting point?
Ordinary Differential Equations is a useful place to start if you want an introduction to independent and dependent variables and derivatives. For a focus on second order equations, the notes on second order linear differential equations begin with a general form and explain what the order means.
What topics do these notes cover?
The notes include introductions to ordinary differential equations, second order linear equations and power series solutions. Other notes cover the method of undetermined coefficients, linearisation, harvesting and the logistic model. You can choose a resource according to the method or model you want to review.
Are the notes free to use?
Yes. Every resource listed here is free to open and study without an account. The notes were made by GoConqr users, so their presentation can vary. A free account also lets you save quiz scores and make your own quizzes, mind maps, flashcards and notes.