Order Differential Equations Dr J Frost jfrosttiffinkingstonschuk Last modified 7 th August 2015 Intro Weve already seen that differential equations are equations which relate ID: 1030907
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1. FP2 Chapter 5 – Second Order Differential EquationsDr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 7th August 2015
2. IntroWe’ve already seen that differential equations are equations which relate and with derivatives. Unsurprisingly, second order differential equations involve the second derivative. 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) Acc Dmp Rst
3. Simple second order differential equationsWe know from the previous chapter that the solution of is .Let’s ‘guess’ that the solution of is similar, and of the form Are the any restrictions on the constants? Let Then and Thus Since thus 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.?????
4. Simple second order differential equationsThe equation is called the auxiliary equation, and if is a root of the auxiliary equation then is a solution of the differential equationWhen the auxiliary equation has two real distinct roots and , the general solution of the differential equation is , where and are arbitrary constants. Find the general solution of the equation Auxiliary equation: General solution is ?
5. Exercise 5A(Just a quickie)Find the general solution of each of the following differential equations: 13579?????
6. Variants: In the previous examples, the auxiliary equation had distinct roots, i.e. . What if we have equal roots? When the auxiliary equation has two equal roots , the general solution is Show that satisfies This is because the root of the auxiliary equation is 3. Bro Side Note: The reason we have to use instead of is similar to why in C4 partial fractions, we have to use if we had a repeated denominator . ?
7. Variants: This is actually exactly the same as when we usually have distinct real roots!Find the general solution of the differential equation Auxiliary equation: General solution is This can be rewritten as (using Chapter 3 knowledge): ???If the auxiliary equation has two imaginary roots , the general solution is where and are arbitrary constants.
8. Variants: So what about more general complex roots ? Find the general solution of the differential equation Auxiliary equation: General solution is This can be rewritten as (using Chapter 3 knowledge): ???If the auxiliary equation has two complex roots , the general solution is where and are arbitrary constants. ?
9. Quickfire Questions!Find solutions to differential equations of the form with the following auxiliary equations (and helpfully provided roots). Auxiliary EquationRootsGeneral SolutionAuxiliary EquationRootsGeneral Solution????????
10. Exercise 5B + 5C
11. Particular IntegralsSo far we’ve always had 0 in the RHS of the differential equation.What if we have some function in terms of ? Solve first. Known as complementary function. (C.F.) Then solve which can be found using appropriate substitution and comparing coefficients. Solution known as particular integral. (P.I.) This is because for the C.F. is 0 and for the P.I., which sum to
12. ExamplesFind the particular integral of the differential equation If is a constant, let the particular integral be a constant too.Subbing in to differential equation: Hence find the general solution of the differential equation Find complementary function:Auxiliary equation: Roots of aux eq: C.F. General solution: ????
13. ExamplesFind the general solution of the differential equation is a linear function, so makes sense to try a linear function for the particular integral.Subbing in:Comparing coefficients:General solution: ??????
14. ExamplesFind the general solution of the differential equation is a quadratic function, so makes sense to try a quadratic function for the particular integral!…General solution: ????
15. ExamplesFind the general solution of the differential equation …General solution: ????Find the general solution of the differential equation …General solution: ????
16. ExamplesBut be warned!!!Your particular integral can’t be part of your complementary function. This is just like how we weren’t allowed to use for the complementary function if the two roots of the auxiliary equation were equal. Find the general solution of the differential equation The difference to when we had is that now the matches the term in the complementary function.Suppose we did use for the particular integral. What goes wrong?Then the general solution might appear to be But is still just an arbitrary constant, so we have exactly the same as the complementary function, which we know gives 0 when subbed into . Thus we end up with . Oh dear!So let This ends up giving So general solution: ????
17. ExamplesBut be warned!!!Your particular integral can’t be part of your complementary function. This is just like how we weren’t allowed to use for the complementary function if the two roots of the auxiliary equation were equal. Find the general solution of the differential equation Auxiliary equation: Complementary function: So particular integral? IT CAN’T BE ! (as there’s a constant term in the c.f.)Instead use General solution: ????So it seems we add a cheeky little in any case where the particular integral is part of the complementary function.
18. Summary So FarForm of Form of particular integralForm of particular integral(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)
19. Test Your UnderstandingFind the value of for which is a particular integral of the differential equation (4 marks)Using your answer to part (a), find the general solution of the differential equation (3 marks) It’s that cheeky little . Why? June 2010 Q8??
20. Test Your UnderstandingFind the general solution of the differential equation June 2012 Q4Be warned: is being used here as was previous used. ?
21. Exercise 5D
22. Boundary ConditionsFind in terms of , given that , and that and at . Sometimes you’re given certain conditions, which allows us to find constants (just as we could with first order differential equations).General solution: Now subbing in boundary conditions into solution:We want to use , so let’s differentiate general solution:Subbing in again:Solving simultaneously:Therefore solution is ??????
23. Test Your Understanding(c) Given that at and , find the particular solution to this differential equation, giving your solution in the form (5)(d) Sketch the curve with equation for (2) June 2010 Q8 (revisited!)?You previously found the general solution in (b) as ?
24. Exercise 5E(Be warned on Q9, Q10 – since it’s , the is what your previously was. Don’t get your variables mixed up!)
25. Using transformationsJust like with first order differential equations, we can use substitutions to turn more complicated second order differential equations into simpler ones.Using the substitution (to obtain an equation involving and ), solve the differential equation Find what is and is so we can substitute them in our differential equation. (chain rule)Since , therefore , thusSimilarly: This is the particularly hard bit.(Note I use a slightly different method to the textbook)(Note: This is different to usual substitutions because here we’re eliminating rather than !)
26. Using transformationsJust like with first order differential equations, we can use substitutions to turn more complicated second order differential equations into simpler ones.Using the substitution (to obtain an equation involving and ), solve the differential equation So we have: Substituting in:General solution: And putting back in: ZOMG Nice.???! To write down:If then:
27. Show that the transformation transforms the equationinto the equation (6)Solve the differentiation equation () to find as a function of . (6)Hence state the general solution of the differential equation () (1) June 2013 Q7Test Your Understanding???
28. 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)
29. SummaryThe official Edexcel specificationNote: Pretty much all questions will involve both a complementary function and a particular integral – it would be too simple for RHS to be 0.