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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.4.35
Capitolo 11, Problema 11.4.35

Differential equations


a. Find a power series for the solution of the following differential equations, subject to the given initial condition
b. Identify the function represented by the power series.


y′(t) − 3y = 10, y(0) = 2

Guida verificata passo dopo passo
1
Rewrite the differential equation in a standard form: \(y'(t) = 3y + 10\).
Assume the solution \(y(t)\) can be expressed as a power series centered at \(t=0\): \(y(t) = \sum_{n=0}^{\infty} a_n t^n\).
Differentiate the power series term-by-term to find \(y'(t) = \sum_{n=1}^{\infty} n a_n t^{n-1}\).
Substitute the series expressions for \(y(t)\) and \(y'(t)\) into the differential equation, and equate coefficients of like powers of \(t\) to form a recurrence relation for the coefficients \(a_n\).
Use the initial condition \(y(0) = 2\) to find \(a_0\), then use the recurrence relation to find subsequent coefficients \(a_n\), thus obtaining the power series solution. Finally, recognize the power series as a known function by comparing it to standard series expansions.

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Power Series Solutions to Differential Equations

A power series solution expresses the solution of a differential equation as an infinite sum of powers of the independent variable. This method is useful when solutions cannot be found using elementary functions. It involves assuming a solution in the form of a series and determining coefficients by substituting into the differential equation.
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Intro to Power Series

Initial Conditions and Their Role

Initial conditions specify the value of the solution and possibly its derivatives at a particular point, allowing for the determination of unknown constants in the general solution. For power series, initial conditions help find the first coefficient(s) and ensure the solution matches the problem's requirements.
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Percorso guidato
05:03
Initial Value Problems

Identifying Functions from Power Series

Once a power series solution is found, it can often be recognized as a known function by comparing it to standard series expansions (e.g., exponential, trigonometric). This identification simplifies understanding the solution's behavior and provides a closed-form expression.
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Representing Functions as Power Series
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Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 11.4 is useful). Use the Maclaurin series


√(1 + x) = 1 + x/2 − x²/8 + x³/16 − ⋯, −1 ≤ x ≤ 1.


√(9 − 9x)

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Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

f(x) = ln √(1 − x²)

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Combining power series Use the power series representation


f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,


to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.


f(3x) = ln (1 − 3x)

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The first three Taylor polynomials for f(x)=√(1+x) centered at 0 are p₀ = 1, p₁ = 1+x/2, and p₂ = 1 + x/2 − x²/8. Find three approximations to √1.1.

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Is ∑ₖ₌₀ ∞ (5x − 20)ᵏ a power series? If so, find the center a of the power series and state a formula for the coefficients cₖ of the power series.

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Use of Tech Linear and quadratic approximation


a. Find the linear approximating polynomial for the following functions centered at the given point a.


b. Find the quadratic approximating polynomial for the following functions centered at a.


c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.


f(x)=e⁻²ˣ, a=0; approximate e⁻⁰ᐧ².

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