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**SECOND-ORDER DIFFERENTIAL EQUATIONS**

PROGRAMME 25 SECOND-ORDER DIFFERENTIAL EQUATIONS

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**Programme 25: Second-order differential equations**

Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

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**Programme 25: Second-order differential equations**

Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

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**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:

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**Programme 25: Second-order differential equations**

Introduction The differential equation: can be re-written to read: that is:

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**Programme 25: Second-order differential equations**

Introduction The differential equation can again be re-written as: where:

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**Programme 25: Second-order differential equations**

Introduction The differential equation: has solution: This means that: That is:

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**Programme 25: Second-order differential equations**

Introduction The differential equation: has solution: where:

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**Programme 25: Second-order differential equations**

Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

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**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:

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**Programme 25: Second-order differential equations**

Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

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**Programme 25: Second-order differential equations**

The auxiliary equation Real and different roots Real and equal roots Complex roots

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**Programme 25: Second-order differential equations**

The auxiliary equation Real and different roots If the auxiliary equation: with solution: where: then the solution to:

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**Programme 25: Second-order differential equations**

The auxiliary equation Real and equal roots If the auxiliary equation: with solution: where: then the solution to:

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**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:

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**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:

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**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::

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**Programme 25: Second-order differential equations**

Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

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**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

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**Programme 25: Second-order differential equations**

Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

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**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.

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**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:

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**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:

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**complementary function + particular integral**

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

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**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:

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**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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