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PROGRAMME 25

PROGRAMME 25. SECOND-ORDER DIFFERENTIAL EQUATIONS. Programme 25: Second-order differential equations. Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations. Programme 25: Second-order differential equations. Introduction Homogeneous equations

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PROGRAMME 25

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  1. PROGRAMME 25 SECOND-ORDER DIFFERENTIAL EQUATIONS

  2. Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

  3. Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

  4. 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. Programme 25: Second-order differential equations Introduction The differential equation: can be re-written to read: that is:

  6. Programme 25: Second-order differential equations Introduction The differential equation can again be re-written as: where:

  7. Programme 25: Second-order differential equations Introduction The differential equation: has solution: This means that: That is:

  8. Programme 25: Second-order differential equations Introduction The differential equation: has solution: where:

  9. Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

  10. 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. Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

  12. Programme 25: Second-order differential equations The auxiliary equation Real and different roots Real and equal roots Complex roots

  13. 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. 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. 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. 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. 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. Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

  19. 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. Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

  21. 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 y1 + y2: • part 1, y1 is the solution to the homogeneous equation and is called the complementary function which is the solution to the homogeneous equation • part 2, y2 is called the particular integral.

  22. Programme 25: Second-order differential equations Inhomogeneous equations Complementary function • Example, to solve: • Complementary function • Auxiliary equation: m2 – 5m + 6 = 0 solution m = 2, 3 • Complementary function y1 = Ae2x + Be3x where:

  23. Programme 25: Second-order differential equations Inhomogeneous equations Particular integral (b) Particular integral Assume a form for y2 as y2 = Cx2 + Dx + E then substitution in: gives: yielding: so that:

  24. Programme 25: Second-order differential equations Inhomogeneous equations Complete solution (c) The complete solution to: consists of: complementary function + particular integral That is:

  25. 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. Programme 25: Second-order differential equations 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

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