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Worked examples and exercises are in the text STROUD PROGRAMME 25 SECOND-ORDER DIFFERENTIAL EQUATIONS.

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Presentation on theme: "Worked examples and exercises are in the text STROUD PROGRAMME 25 SECOND-ORDER DIFFERENTIAL EQUATIONS."— Presentation transcript:

1 Worked examples and exercises are in the text STROUD PROGRAMME 25 SECOND-ORDER DIFFERENTIAL EQUATIONS

2 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

3 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

4 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction For any three numbers a, b and c, the two numbers: are solutions to the quadratic equation: with the properties:

5 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction The differential equation: can be re-written to read: that is:

6 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction The differential equation can again be re-written as: where:

7 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction The differential equation: has solution: This means that: That is:

8 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction The differential equation: has solution: where:

9 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

10 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Homogeneous equations The differential equation: Is a second-order, constant coefficient, linear, homogeneous differential equation. Its solution is found from the solutions to the auxiliary equation: These are:

11 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

12 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations The auxiliary equation Real and different roots Real and equal roots Complex roots

13 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations The auxiliary equation Real and different roots If the auxiliary equation: with solution: where: then the solution to:

14 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations The auxiliary equation Real and equal roots If the auxiliary equation: with solution: where: then the solution to:

15 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations The auxiliary equation Complex roots If the auxiliary equation: with solution: where: Then the solutions to the auxiliary equation are complex conjugates. That is:

16 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations The auxiliary equation Complex roots Complex roots to the auxiliary equation: means that the solution of the differential equation: is of the form:

17 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations The auxiliary equation Complex roots Since: then: The solution to the differential equation whose auxiliary equation has complex roots can be written as::

18 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

19 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Summary Differential equations of the form: Auxiliary equation: Roots real and different: Solution Roots real and the same: Solution Roots complex ( j ): Solution

20 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

21 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Inhomogeneous equations The second-order, constant coefficient, linear, inhomogeneous differential equation is an equation of the type: The solution is in two parts y 1 + y 2 : (a)part 1, y 1 is the solution to the homogeneous equation and is called the complementary function which is the solution to the homogeneous equation (b)part 2, y 2 is called the particular integral.

22 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Inhomogeneous equations Complementary function Example, to solve: (a) Complementary function Auxiliary equation: m 2 – 5m + 6 = 0 solution m = 2, 3 Complementary function y 1 = Ae 2x + Be 3x where:

23 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Inhomogeneous equations Particular integral (b) Particular integral Assume a form for y 2 as y 2 = Cx 2 + Dx + E then substitution in: gives: yielding: so that:

24 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Inhomogeneous equations Complete solution (c) The complete solution to: consists of: complementary function + particular integral That is:

25 Worked examples and exercises are in the text STROUD Programme 25: Second-order differential equations Inhomogeneous equations Particular integrals The general form assumed for the particular integral depends upon the form of the right-hand side of the inhomogeneous equation. The following table can be used as a guide:

26 Worked examples and exercises are in the text STROUD Learning outcomes Use the auxiliary equation to solve certain second-order homogeneous equations Use the complementary function and the particular integral to solve certain second- order inhomogeneous equations Programme 25: Second-order differential equations


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