FP2 Chapter 5 – Second Order Differential Equations
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Intro
Shock absorbers as part of suspension of car subject to force down of car acting under acceleration, and forces up: damping force (proportional to velocity) and restoring force (proportional to extension of spring)
Simple second order differential equations
This is known as an auxiliary equation.
An auxiliary equation is (by definition) is an equation on which the solutions of a differential equation depend.
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Simple second order differential equations
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Exercise 5A
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This is actually exactly the same as when we usually have distinct real roots!
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Quickfire Questions!
Auxiliary Equation | Roots | General Solution |
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Exercise 5B + 5C
Particular Integrals
Examples
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Examples
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Examples
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Examples
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Examples
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Summary So Far
| Form of particular integral |
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(But need to modify if term is same as one of the terms in the complementary function. You will be given this in an exam if the case)
Test Your Understanding
June 2010 Q8
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Test Your Understanding
June 2012 Q4
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Exercise 5D
Boundary Conditions
Sometimes you’re given certain conditions, which allows us to find constants (just as we could with first order differential equations).
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Test Your Understanding
June 2010 Q8 (revisited!)
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Exercise 5E
Using transformations
Just like with first order differential equations, we can use substitutions to turn more complicated second order differential equations into simpler ones.
This is the particularly hard bit.
(Note I use a slightly different method to the textbook)
Using transformations
Just like with first order differential equations, we can use substitutions to turn more complicated second order differential equations into simpler ones.
ZOMG Nice.
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June 2013 Q7
Test Your Understanding
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Exercise 5F
(Note Q7-9 are the type you’re most likely to find in an exam, just like the previous example exam question you did)
Summary
The official Edexcel specification
Note: Pretty much all questions will involve both a complementary function and a particular integral – it would be too simple for RHS to be 0.