Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 11 - Power Series

Problema 11.R.63

Capitolo 11, Problema 11.R.63

A differential equation Find a power series solution of the differential equation y'(x) - 4y + 12 = 0, subject to the condition y(0) = 4. Identify the solution in terms of known functions.

Guida verificata passo dopo passo

Rewrite the differential equation in standard form: \(y'(x) = 4y - 12\).

Assume a power series solution of the form \(y(x) = \sum_{n=0}^{\infty} a_n x^n\) and find its derivative \(y'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1}\).

Substitute the series expressions for \(y(x)\) and \(y'(x)\) into the differential equation to get \(\sum_{n=1}^{\infty} n a_n x^{n-1} = 4 \sum_{n=0}^{\infty} a_n x^n - 12\).

Align powers of \(x\) on both sides by shifting indices as needed, then equate coefficients of like powers of \(x\) to find a recurrence relation for the coefficients \(a_n\).

Use the initial condition \(y(0) = 4\) to find \(a_0\), then solve the recurrence relation to express \(y(x)\) as a power series and recognize it as a known function (likely involving exponentials).

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Power Series Solutions of Differential Equations

A power series solution expresses the unknown function as an infinite sum of powers of the variable, typically centered at a point like x=0. This method is useful when standard solution techniques are difficult, allowing the differential equation to be solved by determining the coefficients of the series.

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Intro to Power Series

Initial Conditions and Their Role

Initial conditions specify the value of the solution or its derivatives at a particular point, enabling the determination of arbitrary constants in the general solution. For example, y(0) = 4 fixes the constant term in the power series, ensuring the solution fits the given problem.

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Percorso guidato

05:03

Initial Value Problems

Solving First-Order Linear Differential Equations

First-order linear differential equations have the form y' + p(x)y = q(x). They can be solved using integrating factors or by recognizing standard solution forms. Identifying the solution in terms of known functions often involves rewriting the equation and integrating accordingly.

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Solving Separable Differential Equations

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