Question

Show that if \(y=\phi(t)\) is a solution of the differential equation \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=g(t)\) where \(g(t)\) is not always zero, then \(y=c \phi(t),\) where \(c\) is any constant other than \(1,\) is not a solution. Explain why this result does not contradict the remark following Theorem \(3.2 .\)

Short answer

Answer: A function of the form \(y=c\phi(t)\), where \(c \neq 1\), cannot be a solution of the given non-homogeneous differential equation because, when we substitute the first and second derivatives of this function into the equation, we obtain a contradiction if \(c \neq 1\). This result does not contradict Theorem 3.2, which deals with homogeneous linear differential equations, because we encountered a non-homogeneous equation in this exercise.

Step-by-step solution

Find the first and second derivatives of \(y=c\phi(t)\)

Let \(y=c\phi(t)\), where \(c\) is a constant other than 1. We can first find the first derivative \(y^{\prime}=c\phi^{\prime}(t)\) and the second derivative \(y^{\prime \prime}=c\phi^{\prime\prime}(t)\).

Substitute the derivatives into the given differential equation

Now we substitute \(y=c\phi(t)\), \(y^{\prime}=c\phi^{\prime}(t)\), and \(y^{\prime \prime}=c\phi^{\prime\prime}(t)\) into the given differential equation: \(c\phi^{\prime\prime}(t) + p(t)(c\phi^{\prime}(t)) + q(t)(c\phi(t)) = g(t)\) We can factor out the constant \(c\) from the equation: \(c(\phi^{\prime\prime}(t) + p(t)\phi^{\prime}(t) + q(t)\phi(t)) = g(t)\) Since \(y=\phi(t)\) is a solution of the differential equation, we know that: \(\phi^{\prime\prime}(t) + p(t)\phi^{\prime}(t) + q(t)\phi(t) = g(t)\)

Compare the equations obtain a contradiction

Now, we compare the two equations: \(c(\phi^{\prime\prime}(t) + p(t)\phi^{\prime}(t) + q(t)\phi(t)) = g(t)\) \(\phi^{\prime\prime}(t) + p(t)\phi^{\prime}(t) + q(t)\phi(t) = g(t)\) If \(c \neq 1\), we have a contradiction because the left-hand side of the first equation can only be equal to \(g(t)\) if \(c=1\). Therefore, the function \(y=c\phi(t)\), where \(c \neq 1\), cannot be a solution of the given non-homogeneous differential equation.

Remarks

This result does not contradict the remark following Theorem 3.2 because Theorem 3.2 deals with homogeneous linear differential equations, where the right-hand side is always zero. In this exercise, we considered a non-homogeneous linear differential equation, where \(g(t)\) is not zero. Hence, Theorem 3.2 does not apply to this situation, and the result does not contradict it.

Key concepts

Non-Homogeneous Linear Differential Equations

Understanding non-homogeneous linear differential equations is fundamental in the study of differential equations. These equations include a function that does not always equal zero on the right side of the equation, known as the non-homogeneity, denoted by \(g(t)\). This distinguishes them from homogeneous equations, where the right side is always zero.

In mathematical terms, a non-homogeneous linear differential equation can be expressed as \(y'' + p(t)y' + q(t)y = g(t)\), where \(p(t)\) and \(q(t)\) are coefficient functions, and \(g(t)\), the non-homogeneity, is a known function that introduces complexity to finding a solution.

The general solution to such an equation consists of two parts: the complementary function (CF), which is the solution of the corresponding homogeneous equation, and the particular solution (PS), which is a solution that includes the effects of \(g(t)\). Fastening our understanding of non-homogeneous linear differential equations allows us to tackle a wide array of problems in physics, engineering, and other applied sciences where forces, sources, or inputs are present.

Second Derivative

The concept of a second derivative in calculus represents the rate of change of the rate of change. This might sound a little confusing at first, but it essentially measures how the slope of a curve changes.

For a function \(y = \text{function of } t\), the second derivative \(y''\) is the derivative of the derivative \(y'\). In the context of differential equations, the second derivative is crucial because it reveals information about the underlying acceleration or concavity of the function in question.

For example, if we have a position function with respect to time, the first derivative corresponds to velocity, and the second derivative corresponds to acceleration. In the case of linear differential equations, the second derivative is typically accompanied by coefficients that may also depend on \(t\), further influencing the behavior of the solution to the differential equation.

Theorem Application

Theorems serve as vital tools that provide proven consensuses within mathematics, aiding in the simplification and resolution of complex problems. When a particular theorem is applied to a problem, it is essential to pay attention to the conditions under which the theorem operates.

Theorem 3.2, often found in textbooks about differential equations, typically pertains to homogeneous linear differential equations and their solutions. It usually states that linear combinations of solutions to homogeneous equations are also solutions. However, this theorem does not directly apply to non-homogeneous equations.

This underlines the importance of recognizing the type of differential equation at hand before applying a theorem. When the given differential equation is non-homogeneous, like in our exercise, we cannot use Theorem 3.2 to claim that any constant multiple of a solution is also a solution. The presence of the non-homogeneous term \(g(t)\) fundamentally changes the situation, illustrating that the application of mathematical theorems requires careful consideration of their scope and limitations.