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). $$ y^{\prime \prime}+e^{t} y^{\prime}+y=0 $$

Short answer

Answer: No, Theorem 8.2(b) cannot be used to guarantee that the given differential equation has a general solution of the form (9) because the coefficient function \(p(t) = e^t\) is not an even function.

Step-by-step solution

Verify Theorem 8.2(b) Conditions

To determine if the given ODE satisfies the conditions of Theorem 8.2(b), the following conditions must hold: 1. The given ODE must be a linear differential equation with coefficients that are continuous functions on the interval \(-\infty < t < \infty\). 2. The functions \(p(t)\) and \(q(t)\) must be even functions. Now let's examine the given ODE: $$ y^{\prime \prime}+e^{t} y^{\prime}+y=0 $$ Our task is to check if the coefficient functions \(p(t) = e^t\) and \(q(t) = 1\) are even functions and continuous on the entire real line.

Examine Continuous Functions

The function \(p(t) = e^t\) is the exponential function, which is continuous for all real values of \(t\). The function \(q(t) = 1\) is a constant function, which is also continuous for all real values of \(t\). Therefore, both \(p(t)\) and \(q(t)\) are continuous functions on the interval \(-\infty < t < \infty\).

Examine Even Functions

Let's check if \(p(t)\) and \(q(t)\) are even functions: 1. For a function to be even, we must have \(f(-t) = f(t)\). For \(p(t) = e^t\), we have \(p(-t) = e^{-t}\) which is not equal to \(e^t\). Therefore, \(p(t) = e^t\) is not an even function. 2. For \(q(t) = 1\), we have \(q(-t) = 1 = q(t)\). Therefore, \(q(t) = 1\) is an even function. Since \(p(t)\) is not even, the given ODE fails to satisfy the conditions of Theorem 8.2(b). Consequently, we cannot use the theorem to represent the general solution in the given form. The answer to the question is no, Theorem \(8.2(\mathrm{~b})\) cannot be used to guarantee that the given differential equation has a general solution of the form (9). The equation fails to satisfy the hypotheses of Theorem \(8.2\) (b) because the coefficient function \(p(t) = e^t\) is not an even function.

Key concepts

Even Functions

In mathematics, an even function is a function that confirms with the property \( f(-t) = f(t) \) for all values of \( t \). This means the function's graph is symmetric with respect to the y-axis. Even functions play an important role in analysis, particularly when examining solutions to differential equations like the one present in our exercise.

To determine whether a function is even, simply substitute \( -t \) for \( t \) in the function’s formula and see if the function remains unchanged. In our given problem, we have \( p(t) = e^t \) and \( q(t) = 1 \). Checking for symmetry:

• \( p(-t) = e^{-t} \), which is not equal to \( e^t \). Therefore, \( p(t) \) is not an even function.
• \( q(-t) = 1 \), which equals \( q(t) \). Hence, \( q(t) \) is an even function.

Unfortunately, since one of our coefficient functions is not even, the specific hypotheses required by Theorem 8.2(b) are not completely satisfied.

Continuous Functions

A continuous function, in a mathematical sense, has no breaks, gaps, or jumps in its graph. This property is crucial in many areas of mathematics and especially in defining certain solutions to differential equations. A function is continuous over some interval if for every point within that interval, the function’s limit matches the function’s value.

Regarding our differential equation, we are looking at the functions \( p(t) = e^t \) and \( q(t) = 1 \). They are assessed as follows:

• \( e^t \): The exponential function \( e^t \) is continuous for all real numbers \( t \), meaning it does not have any discontinuities for \( -\infty < t < \infty \).
• \( 1 \): The function \( q(t) = 1 \) is a constant function and is inherently continuous for all real numbers.

This ensures that, purely from the perspective of continuity, both coefficient functions meet the required condition for the interval of \( -\infty < t < \infty \). But continuity alone is not sufficient without the other hypotheses.

Theorem 8.2 Hypotheses

Theorem 8.2(b) provides criteria for ensuring a differential equation has solutions involving even and odd functions. For the theorem's application, every coefficient function of the differential equation must both be continuous over the entire real line and classify as an even function.

The hypotheses for Theorem 8.2(b) are crucial:

• Continuity of coefficients – As detailed before, the coefficient functions \( p(t) \) and \( q(t) \) must not have any breaks or gaps.
• Evenness – Each coefficient function should satisfy the even function property.

While both of our functions are continuous over the interval specified, only \( q(t) = 1 \) verifies as an even function. \( p(t) = e^t \) does not align with the even function definition, thus failing the conditions required by Theorem 8.2(b). This failure means we cannot use this theorem to easily derive the general solution form prescribed in the problem.

General Solution Form

The general solution of a linear differential equation in the context of Theorem 8.2(b) typically takes the form \( y(t) = c_1 y_e(t) + c_2 y_o(t) \). Here, \( y_e(t) \) and \( y_o(t) \) represent even and odd solutions, respectively, which can be linearly combined to form the general solution.

Applying this to our differential equation requires that the conditions of Theorem 8.2(b) be satisfied so that the solutions \( y_e(t) \) and \( y_o(t) \) can indeed be viewed as even and odd. Since \( p(t) = e^t \) is not an even function, as shown in previous sections, our equation does not comply, implying:

• We lack assurance that the solutions split into even and odd functions.
• The combination form \( y(t) = c_1 y_e(t) + c_2 y_o(t) \) cannot be guaranteed using this theorem.

Thus, while the equation is linear and one part meets the criteria, failing the even function condition of the theorem prevents using this structured form for deriving the general solution.