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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.4.35
Chapter 11, Problem 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

Verified step by step guidance
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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
Related Practice
Textbook Question

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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Textbook Question

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)

Textbook Question

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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Textbook Question

{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.


sin x ≈ x − x³/6 on [π/4, π/4]

Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₁∞ (xᵏ/kᵏ)

Textbook Question

How would you approximate e⁻⁰ᐧ⁶ using the Taylor series for eˣ?

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