Question

Suppose a linear differential equation \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) satisfies the hypotheses of Theorem \(8.2(\mathrm{~b})\), on the interval \(-\infty< t <\infty\). Then, by Exercise 32 , we can assume the general solution of \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) has the form $$ y(t)=c_{1} y_{\mathrm{e}}(t)+c_{2} y_{\mathrm{o}}(t), $$ where \(y_{\mathrm{e}}(t)\) is an even solution of \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) and \(y_{\mathrm{o}}(t)\) is an odd solution. In each of the following exercises, determine whether Theorem \(8.2(\mathrm{~b})\) can be used to guarantee that the given differential equation has a general solution of the form (9). If your answer is no, explain why the equation fails to satisfy the hypotheses of Theorem \(8.2\) (b). $$ y^{\prime \prime}+y^{\prime}+t^{2} y=0 $$

Short answer

Answer: Yes, the general solution can be written in the given form, as the differential equation satisfies the conditions for Theorem 8.2(b).

Step-by-step solution

Identifying the given equation and notation

The given linear differential equation is: $$y'' + y' + t^2 y = 0$$ Let's denote the coefficients as follows: \(p(t) = 1\) \(q(t) = t^2\)

Verifying conditions of Theorem 8.2(b)

To apply Theorem 8.2(b), we need to check if the given equation satisfies the following conditions: 1. \(p(t)\) and \(q(t)\) are continuous on the interval \((-\infty, \infty)\). 2. Both \(p(t)\) and \(q(t)\) are even functions. Let's check these conditions one by one: 1. \(p(t) = 1\) is continuous on the interval \((-\infty, \infty)\). \(q(t) = t^2\) is continuous on the interval \((-\infty, \infty)\). 2. Since \(p(t) = 1\) and does not depend on \(t\), it is an even function. As for \(q(t) = t^2\), it is an even function, because for every \(t\), we have \(q(-t) = (-t)^2 = t^2 = q(t)\). Both conditions are satisfied, so we can use Theorem 8.2(b) to find the general solution in the given form.

Finding the even and odd solutions

According to Theorem 8.2(b), the general solution for the given equation has the form: $$y(t) = c_1 y_{\text{e}}(t) + c_2 y_{\text{o}}(t)$$ However, the question asks only if the general solution can be in the form of given function and it does not provide specific even or odd solutions to determine \(y_{\text{e}}(t)\) and \(y_{\text{o}}(t)\). In conclusion, since the given differential equation satisfies the conditions for Theorem 8.2(b), its general solution can be in the form of \(y(t) = c_1 y_{\text{e}}(t) + c_2 y_{\text{o}}(t)\).

Key concepts

Theorem 8.2(b)

Theorem 8.2(b) is a significant statement within the realm of differential equations, and understanding it is crucial for solving certain types of problems. This theorem indicates conditions under which a linear differential equation with continuous coefficients has solutions that can be expressed as a combination of even and odd functions. Specifically, if the coefficients of the linear differential equation are even functions and continuous on the entire real line, then there exists a general solution composed of an even function and an odd function.

The impact of Theorem 8.2(b) is profound because it not only ensures the existence of a solution but also provides a structured form for the general solution. This shape of the solution simplifies the process of solving differential equations by narrowing down the potential forms that the solution can take, thereby making it easier for students and mathematicians to tackle complex problems.

Even and Odd Functions

In mathematics, functions can exhibit symmetry, which is where the concepts of even and odd functions come into play. An even function is defined by the property that its graph is symmetric with respect to the y-axis, which gives us the algebraic condition that for every input 't', the function value does not change when 't' is replaced with '-t'. In simpler terms, an even function satisfies the equation \( f(-t) = f(t) \).

On the other hand, an odd function has a graph that exhibits point symmetry about the origin. In this case, flipping the input 't' to '-t' flips the sign of the function's output, expressed algebraically as \( f(-t) = -f(t) \). Recognizing whether the solutions to a differential equation are even or odd can be critical in finding the general solution, especially when applying Theorem 8.2(b). The existence of even and odd functions within the general solution reflects the original equation's symmetry and provides a powerful tool for solving these equations analytically.

General Solution of Differential Equations

The general solution of a differential equation represents the most comprehensive expression that includes all possible specific solutions to the equation. It is a family of functions, typically involving arbitrary constants that accommodate for all the initial conditions that could be applied to the equation. When dealing with second-order linear differential equations, as seen in Theorem 8.2(b), the general solution can often be expressed as a sum of two functions, each multiplied by a constant.

In practice, determining the general solution involves finding the functions that satisfy the differential equation given any initial condition within a specified domain. Theorem 8.2(b)'s approach of using a linear combination of an even and an odd function greatly simplifies the process of finding these solutions by limiting the form the solutions can take. It underscores the importance of understanding the characteristics of the equation in order to apply the right methods for solving it.

Continuity of Functions

Continuity is a fundamental concept in calculus that describes a function's behavior around certain points. A function is said to be continuous at a point if a small change in the input results in a small change in the output. More formally, a function \( f(t) \) is continuous at a point \( t = a \) if the left-hand limit, the right-hand limit, and the value of the function at 'a' all exist and are equal. In the context of Theorem 8.2(b), the continuous nature of the coefficients in a linear differential equation means that there are no abrupt changes or breaks in the behavior of the equation across the interval in question.

Continuity plays a critical role in applying Theorem 8.2(b) because the theorem's conditions require the coefficients of the differential equation to be continuous on the entire real line. This guarantees that the fundamental properties of the solutions, like their smoothness and the predictability of their behavior, remain intact across the domain, thus enabling the identification of general solutions of specific forms, as with even and odd functions.