Question

A function is said to be periodic with period \(p\) if $$ f(x+n p)=f(x) $$ where \(n\) is an integer. Suppose that \(f(x)\) is continuous and periodic with period \(p\) for all \(x\). Show that if \(\phi(x)\) is a solution of the homogeneous equation $$ y^{\prime}+f(x) y=0 $$ then \(\phi(x+p)\) is also a solution. Show that for some constant \(c\), $$ \phi(x+p)=c \phi(x) $$ for all \(x\).

Short answer

**Answer:** There exists a constant \(c\) such that \(\phi(x+p)=c\phi(x)\) for all \(x\).

Step-by-step solution

Show that \(\phi(x+p)\) is a solution of the homogeneous equation.

We are given the homogeneous equation: $$ y^{\prime}+f(x)y=0 $$ We know that \(\phi(x)\) is a solution to this equation. Let's find the derivative of \(\phi(x+p)\) with respect to \(x\): $$ \frac{d}{dx}(\phi(x+p)) = \phi^{\prime}(x+p) $$ Now, let's substitute \(y = \phi(x+p)\) into the given homogeneous equation and see if it satisfies the equation: $$ \phi^{\prime}(x+p)+f(x)\phi(x+p) \stackrel{?}{=} 0 $$ Since \(f(x)\) is periodic with period \(p\), we have \(f(x+p)=f(x)\). Therefore, $$ \phi^{\prime}(x+p)+f(x+p)\phi(x+p) = \phi^{\prime}(x+p)+f(x)\phi(x+p) = 0 $$ So, \(\phi(x+p)\) is indeed a solution to the given homogeneous equation.

Show that there exists a constant \(c\) such that \(\phi(x+p)=c\phi(x)\) for all \(x\).

To find the constant \(c\), first, observe that since \(f(x)\) is continuous and periodic with period \(p\), we can write \(\phi(x)\) as: $$ \phi(x) = Ce^{-\int_{0}^{x}f(t) dt} $$ where \(C\) is an integration constant. Now, let's find the expression for \(\phi(x+p)\) using the above formula: $$ \phi(x+p) = Ce^{-\int_{0}^{x+p}f(t) dt} $$ We can break the integral into two parts: $$ \phi(x+p) = Ce^{-\left(\int_{0}^{x}f(t) dt + \int_{x}^{x+p}f(t) dt\right)} $$ Since \(f(x)\) is periodic with period \(p\), the integral of \(f(t)\) over one period is a constant, say \(K\): $$ \int_{x}^{x+p}f(t) dt = K $$ Now, we have $$ \phi(x+p) = Ce^{-\left(\int_{0}^{x}f(t) dt + K\right)} = Ce^{-\int_{0}^{x}f(t) dt} \cdot Ce^{-K} $$ Comparing this expression with the expression for \(\phi(x)\), we find the constant \(c\): $$ c = Ce^{-K} $$ Thus, we have shown that there exists a constant \(c\) such that \(\phi(x+p)=c\phi(x)\) for all \(x\).

Key concepts

Periodic Functions

Periodic functions are fascinating mathematical objects that repeat their values at regular intervals. If a function \( f(x) \) is periodic with period \( p \), it means that after every \( p \) units along the x-axis, the function value repeats itself: \( f(x + np) = f(x) \) for any integer \( n \). This property makes periodic functions useful in modeling phenomena that naturally repeat, like sound waves or seasons. These functions are fundamental in calculus and differential equations because they introduce a regular pattern that can simplify the analysis of complex systems. For instance, the trigonometric functions \( \sin(x) \) and \( \cos(x) \) are classic examples of periodic functions with period \( 2\pi \). Understanding periodicity is crucial in solving differential equations where the repeating nature of the function can aid in finding solutions.

Homogeneous Equations

In the study of differential equations, a homogeneous equation has the form \( y' + f(x)y = 0 \). The term 'homogeneous' might suggest uniformity or equality, although in this context it means that the equation is equal to zero. Such equations have solutions that can be scaled by a constant factor. For example, if \( \phi(x) \) is a solution, then \( c\phi(x) \), where \( c \) is a constant, is also a solution.
This property is demonstrated in the concept of superposition, which is key in linear equations. Finding solutions to homogeneous equations often involves recognizing structures like periodicity and using them to simplify the problem further.
The step-by-step solution of the problem shows that if \( \phi(x) \) solves the homogeneous equation, then \( \phi(x+p) \), being a shifted form of \( \phi(x) \), must also be a solution due to the periodic nature of \( f(x) \). This emphasizes how math properties interplay to reveal deeper patterns in solutions.

Continuity

Continuity is an essential idea in calculus and analysis, referring to functions that change smoothly without breaks or jumps. A continuous function is one where small changes in the input lead to small changes in the output, meaning you can draw its graph without lifting your pen.
In the context of differential equations, continuity ensures that the behavior of solutions is predictable and well-behaved. It implies certain integrals can be computed reliably. For periodic functions like \( f(x) \), continuity means their cyclical nature doesn't introduce sharp turns or discontinuities, simplifying the work of finding and verifying solutions.
During the exercise solution, continuity of \( f(x) \) facilitates proving periodic properties of solutions such as \( \phi(x+p) = c\phi(x) \), allowing transformations and shifts without loss of generality.

Integration

Integration is a mathematical way to accumulate quantities, often realized as finding the area under a curve. It is the inverse operation of differentiation. In solving differential equations, integration allows us to determine solutions through initial and boundary conditions.
In this solution, integration is used to express solutions of the homogeneous differential equation. Particularly with periodic functions, it can sometimes simplify complex expressions by taking advantage of repetitive patterns.
The exercise leverages integration to show that \( \phi(x) = Ce^{-\int_0^x f(t) dt} \), and further manipulates this expression to demonstrate the constant relationship \( \phi(x+p) = c\phi(x) \). The periodicity and continuity of \( f(x) \) play critical roles in this integration process, showing how different mathematical concepts align to solve differential equations.