Differential Equations - Second-Order Equations - Quick Guides

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Study GuideDifferential Equations–Second-Order Equations1. Second-Order Homogeneous Equations1.1What Does “Homogeneous” Mean Here?The termhomogeneous differential equationis used in two different ways, depending on context.First meaning (earlier topic)Afirst-orderequation of the formis called homogeneous if(M)and(N)are homogeneous functions of the same degree.Second meaning (used much more often)A differential equation ofany orderis calledhomogeneousifall terms involving the unknownfunction are on one side, and the other side is zero.For example:•y''-2y' + y = 0 (homogeneous)•y''-2y' + y = x (not homogeneous)This second definition is the one we will focus on in this chapter.1.2Homogeneous vs Nonhomogeneous EquationsA general second-ordernonhomogeneouslinear equation looks likeIf we replace the right-hand side by zero, we get thecorresponding homogeneous equation:The equation (**) plays a crucial role in solving (*).

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Study Guide1.3Two Fundamental TheoremsThese two results explain how solutions of homogeneous and nonhomogeneous equations arerelated.Theorem AIf(y1(x))and(y2(x))arelinearly independent solutionsof a linear homogeneous equation, thenevery solutionof that equation can be written as a linear combination of them:This expression is called thegeneral solutionof the homogeneous equation.Theorem BIf:•(yp(x))isany particular solutionof the nonhomogeneous equation, and•(yh(x))is thegeneral solution of the corresponding homogeneous equation,then the general solution of the nonhomogeneous equation isIn words:General solution of a nonhomogeneous equation=general solution of the corresponding homogeneous equation•any particular solutionThe function(yh)is also called thecomplementary function.Example 1:General Solution of a Homogeneous EquationConsider:We are told that

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Study Guideare solutions.Step 1: Form Linear CombinationsBy Theorem A, every solution has the form:Step 2: (Optional) Verify the SolutionDifferentiate:Substituting into the equation shows thatis satisfied.Final AnswerExample 2:Solving a Nonhomogeneous EquationVerify that(y = 4x-5)satisfies

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Study GuideStep 1: Verify the Particular SolutionIf(y = 4x-5), then:Substitute:So(y = 4x-5)isa particular solution.Step 2: Solve the Corresponding Homogeneous EquationThe homogeneous equation is:We are given two linearly independent solutions:So,Step 3: Apply Theorem BAdd the particular solution:Example 3:Trigonometric SolutionsFirst, verify that:solve the homogeneous equation:Both satisfy the equation, and they are linearly independent.

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Study GuideStep 1: Write the Homogeneous SolutionBy Theorem A:Step 2: Solve the Nonhomogeneous EquationNow consider:By inspection, you can see that(y = x)is a particular solution.Step 3: Write the General SolutionUsing Theorem B:Summary•A homogeneous equation haszero on the right-hand side•Solutions of homogeneous equations combinelinearly•Linearly independent solutions form thebuilding blocksof all solutions•For nonhomogeneous equations:oFirst solve the corresponding homogeneous equationoThen addanyparticular solution•These ideas apply tolinear equations of any order

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Study Guide2. The Method of Undetermined Coefficients2.1Why This Method ExistsTo solve anonhomogeneous linear differential equation, we learned (from Theorem B) that:General solution = homogeneous solution + particular solutionSo the real challenge is often finding aparticular solution.When the nonhomogeneous term has a special form, we can use a powerful and systematictechnique called themethod of undetermined coefficients.2.2When Can This Method Be Used?Consider the nonhomogeneous equationThe method of undetermined coefficients worksonly whenthe nonhomogeneous term(d(x))belongsto a special class of functions called those with afinite family of derivatives.2.3What Is a “Finite Family of Derivatives”?A function has afinite family of derivativesif all of its derivatives can be written usingonly a finitenumber of functions.Example:A Function With a Finite FamilyLetIts derivatives cycle:•d' = cos x•d'' =-sin x•d''' =-cos x•d4= sin xAll derivatives involve only sin x and cos x.

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Study GuideSo thefamily of derivativesis:Example:A Function Without a Finite FamilyLetIts derivatives keep producing higher and higher powers of sec x and tan x.There isno finite setthat contains all derivatives.Conclusion:The method of undetermined coefficientscannotbe used when d(x) = tan x2.4Functions That Work with This MethodThe following functionsdohave finite derivative families and are suitable for this method:2.5Linear CombinationsAlinear combinationof functions(y1, y2,…….yn)is any expression of the formwhere the(ci)are constants.

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Study GuideKey Idea of the MethodTo find a particular solution, form themost general linear combinationof the functions in the familyof(d(x)), then substitute it into the differential equation and solve for the constants.Example 1:Identifying a FamilyIfThe family is:(The constant 5 is ignored.)Example 2:Product of FunctionsIf•Family of(x):({x, 1})•Family of (sin 2x):(sin 2x, cos 2x)So the combined family is:Example 3:Polynomial Forcing TermFind a particular solution ofStep 1: Choose a Trial FunctionThe family of(5x2)is(x2, x, 1), so try:

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Study GuideStep 2: Substitute and Match CoefficientsSubstituting into the equation and combining like terms gives:Equating coefficients:Solving:2.6Particular SolutionExample 4:Trigonometric Forcing TermSolve:Step 1: Choose Trial FunctionFamily of sin x: sin x, cos xTry:

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Study GuideStep 2: Substitute and SolveAfter substitution:So:This gives:Particular SolutionComplete SolutionExample 5:Exponential Forcing TermSolve:Try:Substitution gives:

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