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.

Verified step by step guidance

1. Start by understanding the differential equation y''(t) + y(t) = 0, where y''(t) is the second derivative of y with respect to t.
2. 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.
3. 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).
4. 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).
5. 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.

Key Concepts

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.

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.

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.