Question

In each exercise, assume that \(y(t)=C_{1} \sin \omega t+C_{2} \cos \omega t\) is the general solution of \(y^{\prime \prime}+\omega^{2} y=0\). Find the unique solution of the given initial value problem. $$y^{\prime \prime}+y=0, \quad y(\pi)=0, \quad y^{\prime}(\pi)=2$$

Short answer

Answer: The unique solution of the initial value problem is \(y(t) = -2 \sin t\).

Step-by-step solution

Identify the general solution of the given differential equation

The general solution of the differential equation $$y'' + y = 0$$ is given by $$y(t)=C_{1} \sin \omega t+C_{2} \cos \omega t$$ with \(\omega = 1\) (because \(\omega^2 = 1\)). So, we have $$y(t)=C_{1} \sin t+C_{2} \cos t$$.

Derive the general solution to find \(y'(t)\)

Differentiate the general solution with respect to \(t\): $$y'(t) = C_1 \cos t - C_2 \sin t.$$

Apply the initial conditions to find \(C_1\) and \(C_2\)

Use the first initial condition, \(y(\pi) = 0\): $$0 = C_1 \sin \pi + C_2 \cos \pi = -C_2.$$ Thus, \(C_2 = 0\). Now use the second initial condition, \(y'(\pi) = 2\): $$2 = C_1 \cos \pi - 0 \cdot \sin \pi = -C_1.$$ Thus, \(C_1 = -2\).

Write the unique solution

Plug the values of \(C_1\) and \(C_2\) into the general solution: $$y(t) = -2 \sin t + 0 \cdot \cos t = -2 \sin t.$$ This is the unique solution of the given initial value problem.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In the exercise, we deal with the second-order differential equation \(y'' + y = 0\). This equation tells us how the second derivative of \(y\) is related to \(y\) itself. These types of equations are essential in describing various physical phenomena, such as motion, heat, and waves. To solve such an equation, we look for the function \(y(t)\) that satisfies this relationship.

The given differential equation is homogeneous, meaning there's no external forcing term (like an equal sign with a function that's not zero). That makes it somewhat simpler to solve, as the focus is on finding the behavior dictated solely by the initial conditions and inherent dynamics described by the equation.

General Solution

The general solution to a differential equation is a family of functions that contains all possible solutions. For the differential equation \(y'' + y = 0\), the general solution is \(y(t) = C_1 \sin t + C_2 \cos t\).

The coefficients \(C_1\) and \(C_2\) are determined based on the initial conditions provided in the problem. This general solution involves trigonometric functions because the differential equation resembles a harmonic oscillator, a system that naturally integrates sine and cosine functions as solutions. These functions capture the repetitive, oscillating nature of problems like these.

Initial Conditions

Initial conditions specify the state of the system at a particular time, letting us determine the unknowns in the general solution to find a unique solution. In the exercise, we have \(y(\pi) = 0\) and \(y'(\pi) = 2\). Initial conditions are critical because they ensure that we only get one specific solution that fits the exact setup we're looking at.

By plugging these initial conditions into the general solution and its derivative, we solve for the constants \(C_1\) and \(C_2\). In this problem, they turn out to be \(C_1 = -2\) and \(C_2 = 0\), which provides us with the unique solution \(y(t) = -2 \sin t\).

Trigonometric Functions

Trigonometric functions like sine and cosine are vital in solving differential equations, especially those models involving periodic and oscillatory behavior. The function \(y(t) = C_1 \sin t + C_2 \cos t\) uses these functions to describe the solution to the differential equation \(y'' + y = 0\).

Sine and cosine are chosen because they naturally arise in the solutions to equations with periodic coefficients. Their properties, such as having derivatives that are shifts of themselves, make them uniquely suited to describe systems where you expect solutions to repeat at regular intervals (like simple harmonic motion). In our result, the unique solution \(y(t) = -2 \sin t\) illustrates motion akin to waves or oscillations, depicting the influence of trigonometric functions in modeling such systems.