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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.5.79.b
Chapter 3, Problem 3.5.79.b

A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.
b. Show that y = B cos t satisfies the equation for any constant B.

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Start by understanding the differential equation y''(t) + y(t) = 0, where y''(t) is the second derivative of y with respect to t.
Consider the proposed solution y(t) = B cos(t), where B is a constant. We need to verify that this function satisfies the differential equation.
Calculate the first derivative of y(t) = B cos(t) with respect to t. The derivative of cos(t) is -sin(t), so y'(t) = -B sin(t).
Calculate the second derivative of y(t) = B cos(t). The derivative of -B sin(t) is -B cos(t), so y''(t) = -B cos(t).
Substitute y(t) = B cos(t) and y''(t) = -B cos(t) into the differential equation y''(t) + y(t) = 0. This gives -B cos(t) + B cos(t) = 0, which simplifies to 0 = 0, confirming that y = B cos(t) satisfies the equation for any constant B.

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Key Concepts

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

Differential Equations

A differential equation is a mathematical equation that relates a function to its derivatives. It describes how a quantity changes over time or space, and can be classified into ordinary differential equations (ODEs) and partial differential equations (PDEs) based on the number of independent variables involved.
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Second-Order Derivatives

In the context of differential equations, a second-order derivative refers to the derivative of a derivative, indicating how the rate of change of a function itself changes. For example, in the equation y''(t), the notation signifies the second derivative of the function y with respect to the variable t, which is crucial for analyzing the behavior of the function.
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Higher Order Derivatives

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental periodic functions that arise in various mathematical contexts, including solutions to differential equations. The function y = B cos(t) represents a cosine wave, where B is a constant that affects the amplitude, and it is often used to express solutions to second-order linear differential equations with constant coefficients.
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Introduction to Trigonometric Functions
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