Differential Equations - Power Series - Quick Guides

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Study GuideDifferential Equations–Power Series1. Solutions of Differential EquationsIn this section, we learn howpower seriescan be used to solve differential equations. This method isespecially useful when standard techniques don’t work or when solutions cannot be written usingfamiliar elementary functions.We’ll start withfirst-order equations, then move on tosecond-order equations, using severalexamples along the way.1.1Power Series Solutions for First-Order EquationsA key fact from calculus is thatpower series can be differentiated term by termwithin their intervalof convergence. Because of this, we can try solving a differential equation byassumingthe solutionhas the formThe idea is simple:1.Assume a power series solution.2.Differentiate the series as needed.3.Substitute into the differential equation.4.Match coefficients to find arecurrence relationfor the coefficients(cn).Example 1:Solving(y'-xy = 0)Step 1: Assume a power series solutionDifferentiate:

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Study GuideStep 2: Substitute into the equationSubstituting into(y'-xy = 0)gives:Write out the first few terms and combine like powers of(x). This leads to an equation whereeachcoefficient must equal zero.Step 3: Find the recurrence relationFrom the pattern, we obtain:•(c1= 0)•For(n ≥ 2):This recurrence relation determines all coefficients from(c0).Step 4: Write the solutionAll coefficients with odd indices are zero. The solution becomes:This series can be written compactly as:Step 5: Recognize the functionThis power series is the expansion of. So the solution is:This confirms that the power series method agrees with the exact solution.

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Study GuideExample 2:Solving an Initial Value Problem]Step 1: Assume a power seriesSubstitute into the equation and combine like terms.Step 2: Match coefficientsThis produces the relations:•(c1= c0)•(c2= (1/2)(1 + c0))•For(n ≥ 3):Step 3: Apply the initial conditionUsing(y(0) = 1), we get:Substituting back gives the solution:Step 4: Express in closed form (optional)This series simplifies to:You can verify that this function satisfies both the differential equation and the initial condition.

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Study Guide1.2Moving to Second-Order EquationsPower series methods also work forsecond-order linear differential equations, but the process ismore involved.Any homogeneous second-order linear equation can be written as:1.3Ordinary Points vs. Singular PointsLet(x0)be a point of interest.•If both(p(x))and(q(x))areanalyticat(x0), then(x0)is called anordinary point.•If either function is not analytic at(x0), then(x0)is asingular point.Because the power series method is much simpler at ordinary points, we focus on those.Example 3:A Second-Order IVPStep 1: Assume a power seriesSubstitute(y),(y'), and(y'')into the equation.Step 2: Re-index and combine seriesTo combine terms, all series must involve the same power of(x). This requiresre-indexingsomesums.Once combined, the equation becomes a single power series equal to zero.

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Study GuideStep 3: Set coefficients equal to zeroThis produces:•A condition relating(c2)and(c0)•A recurrence relation for(c(n+2))in terms of(cn)Step 4: Apply initial conditionsFrom:•(y(0) = 2 = c0= 2)•(y'(0) = 3 = c1= 3)All remaining coefficients follow from the recurrence relation.The final solution is:Example 4:Power Series Solution ofIn this example, we’ll find apower series solution centered at(x = 0). The process is longer thanearlier examples, but the steps are the same—just more algebra.Step 1: Assume a power series solutionAs usual, we start by assuming the solution has the formFrom this, we compute the derivatives:

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Study GuideStep 2: Substitute into the differential equationSubstitute(y),(y'), and(y'')intoAfter substitution, the equation becomes a sum of several power series involving different powers of(x).At this stage, the expression looks complicated—but that’s normal.Step 3: Rewrite everything in terms of powers of(xn)To combine all terms intoone single summation, every series must involve the same power of(x).This requiresre-indexingsome of the sums so that each one is written in terms of(xn). Once this isdone, all terms can be combined into a single power series.Step 4: Combine into a single seriesAfter re-indexing and simplifying, the equation takes the form:Step 5: Set coefficients equal to zeroFor this equation to hold forall values of(x), every coefficient must be zero.This gives:•(c1+ 2c2= 0)•(2c2+ 6c3= 0)and for(n ≥ 2), the recurrence relation:

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