Programme 26 Secondorder differential equations PROGRAMME 26 SECONDORDER

Programme 26: Second-order differential equations PROGRAMME 26 SECOND-ORDER DIFFERENTIAL EQUATIONS STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: 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: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction The differential equation: can be re-written to read: that is: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction The differential equation can again be re-written as: where: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction The differential equation: has solution: This means that: That is: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction The differential equation: has solution: where: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: 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: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations The auxiliary equation Real and different roots Real and equal roots Complex roots STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations The auxiliary equation Real and different roots If the auxiliary equation: with solution: where: then the solution to: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations The auxiliary equation Real and equal roots If the auxiliary equation: with solution: where: then the solution to: STROUD Worked examples and exercises are in the text

Programme 26: 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: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations The auxiliary equation Complex roots to the auxiliary equation: means that the solution of the differential equation: is of the form: STROUD Worked examples and exercises are in the text

Programme 26: 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: : STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: 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 (α + jβ): Solution STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: 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. STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Inhomogeneous equations Complementary function Example, to solve: (a) Complementary function Auxiliary equation: m 2 – 5 m + 6 = 0 solution m = 2, 3 Complementary function y 1 = Ae 2 x + Be 3 x where: STROUD Worked examples and exercises are in the text

Programme 26: 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: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Inhomogeneous equations Complete solution (c) The complete solution to: consists of: complementary function + particular integral That is: STROUD Worked examples and exercises are in the text

Programme 26: 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: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations Particular solution STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Particular solution We have just seen that the complete solution to: is: Which contains the two constants, A and B. These two constants are arbitrary because whatever values are chosen for them, when inserted into the above equation for y it will still be a solution to the differential equation. This means, of course, that there is an infinite number of solutions to the differential equation, each one having specific values for A and B. STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Particular solution To select just one particular solution requires additional information and this information is provided by what are called boundary conditions that take the form of a given specific value of y and its derivative for a particular value of x. For example, if we were to impose the boundary conditions that at: we would find that: STROUD Worked examples and exercises are in the text

Programme 26: Second-order differential equations Learning outcomes üUse the auxiliary equation to solve certain second-order homogeneous equations üUse the complementary function and the particular integral to solve certain secondorder inhomogeneous equations üImpose boundary conditions to find a particular solution STROUD Worked examples and exercises are in the text