Question

Study the behavior of the solutions of the initial value problem: $$ u^{\prime \prime}(x)+\lambda u(x)=0, \quad u(0)=0, \quad u^{\prime}(0)=1 $$ as \(\lambda\) increases from 0 to \(\infty\); note in particular the appearance of values of \(\lambda\) for which the condition \(u(1)=0\) is satisfied. It can be shown that the same qualitative picture holds for the general Sturm- Liouville problem of Section 10.13.

Short answer

Short Answer: The values of \(\lambda\) that satisfy the condition \(u(1)=0\) for the given initial value problem are: $$ \lambda=(n\pi)^2, \quad n\in\mathbb{Z}, \; n>0 $$ The solution exhibits oscillatory behavior for these values of \(\lambda\) and correspond to the eigenvalues for the Sturm-Liouville problem.

Step-by-step solution

Consider the Types of Solutions for Different Values of \(\lambda\)

Depending on the value of \(\lambda\), the general solution to the differential equation will take on different forms. If \(\lambda>0\), the general solution is a linear combination of sine and cosine functions; if \(\lambda=0\), the general solution is a linear function; and if \(\lambda<0\), the general solution is a linear combination of hyperbolic sine and cosine functions. Therefore, we will analyze each case separately.

Case \(\lambda > 0\) (Positive \(\lambda\))

For \(\lambda>0\), the general solution is given by: $$ u(x)=A\cos(\sqrt{\lambda}x)+B\sin(\sqrt{\lambda}x) $$ Now, we apply the initial conditions: \(u(0)=0\) and \(u^{\prime}(0)=1\). This gives us: $$ u(0)=A\cos(0)+B\sin(0)=0 \implies A=0 $$ $$ u^{\prime}(x)=-A\sqrt{\lambda}\sin(\sqrt{\lambda}x)+B\sqrt{\lambda}\cos(\sqrt{\lambda}x) $$ $$ u^{\prime}(0)=-A\sqrt{\lambda}\sin(0)+B\sqrt{\lambda}\cos(0)=1 \implies B=\frac{1}{\sqrt{\lambda}} $$ So the solution in this case is: $$ u(x)=\frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda}x) $$ Now, we want to find the values of \(\lambda\) such that \(u(1)=0\): $$ \frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda})=0 \implies \sin(\sqrt{\lambda})=0 $$ Since the sine function is zero at integer multiples of \(\pi\), we get: $$ \sqrt{\lambda}=n\pi, \quad n\in\mathbb{Z} $$ And then we find the positive \(\lambda\) values as: $$ \lambda = (n\pi)^2 $$

Case \(\lambda = 0\)

For \(\lambda=0\), the general solution is given by: $$ u(x)=Ax+B $$ Applying the initial conditions \(u(0)=0\) and \(u^{\prime}(0)=1\), we get: $$ u(0)=A(0)+B=0 \implies B=0 $$ $$ u^{\prime}(x)=A $$ $$ u^{\prime}(0)=A=1 \implies A=1 $$ So the solution in this case is: $$ u(x)=x $$ However, for this case, the condition \(u(1)=0\) is not satisfied.

Case \(\lambda < 0\) (Negative \(\lambda\))

For \(\lambda<0\), let \(\mu=-\lambda>0\). The general solution is then given by: $$ u(x)=A\cosh(\sqrt{\mu}x)+B\sinh(\sqrt{\mu}x) $$ Applying the initial conditions \(u(0)=0\) and \(u^{\prime}(0)=1\), we get: $$ u(0)=A\cosh(0)+B\sinh(0)=0 \implies A=0 $$ $$ u^{\prime}(x)=B\sqrt{\mu}\cosh(\sqrt{\mu}x) $$ $$ u^{\prime}(0)=B\sqrt{\mu}\cosh(0)=1 \implies B=\frac{1}{\sqrt{\mu}} $$ So the solution in this case is: $$ u(x)=\frac{1}{\sqrt{\mu}}\sinh(\sqrt{\mu}x) $$ However, since \(\sinh\) is never zero for \(x>0\), there are no values of \(\lambda\) in this case that satisfy the condition \(u(1)=0\).

Summary

In conclusion, the values of \(\lambda\) that satisfy the condition \(u(1)=0\) are given by: $$ \lambda=(n\pi)^2, \quad n\in\mathbb{Z}, \; n>0 $$ The solution exhibits oscillatory behavior for these values of \(\lambda\), as indicated by the sine function in the solution for the positive \(\lambda\) case. These values of \(\lambda\) correspond to the eigenvalues for this Sturm-Liouville problem. The same qualitative picture holds for the general Sturm-Liouville problem as described in Section 10.13.

Key concepts

Understanding Differential Equations

Differential equations are fundamental in mathematics and science, dealing with the rate of change measured by derivatives. These equations describe the relationships between functions and their derivatives, often representing physical phenomena such as motion, heat, or waves.

In our exercise, we tackled a second-order linear homogeneous differential equation. The equation is 'homogeneous' meaning the only terms it includes are those with the function being solved for or its derivatives, and 'linear' because the function and its derivatives appear only to the first power, without any multiplication between them. Such equations often emerge in the study of physical systems, and their solutions can represent a wide range of behaviors, from simple motion to complex oscillations.

In solving such an equation, it is crucial to understand initial conditions, as they determine the specific solution to our problem from a family of potential solutions. In our case, these conditions are given for the function and its derivative at the point x=0. The behavior of solutions changes significantly with different parameters, with \( u(x) \) shaped by the value of \( \lambda \) in the equation.

The Role of Eigenvalues

Eigenvalues are pivotal in mathematics, physics, and engineering, particularly in the context of linear transformations and stability analysis. They appear as critical parameters that determine the nature of solutions to certain types of equations, such as the Sturm-Liouville problem we're examining.

In the Sturm-Liouville problem, eigenvalues emerge when we seek values of the parameter \( \lambda \) for which the differential equation has non-trivial solutions that meet specific boundary conditions. These solutions, corresponding to the eigenvalues, are also known as 'eigenfunctions'. The existence of these eigenvalues is profound as they articulate the resonant frequencies of a system, without external forces being applied, manifesting natural patterns of vibration.

From our step-by-step solution, we see that the eigenvalues for our problem are \( \lambda = (n\pi)^2 \), where \( n \) is a positive integer. These discrete values lead to solutions that articulate how the system will behave, oscillate, and respond when 'resonated' at certain frequencies.

Investigating the Initial Value Problem

An initial value problem in the field of differential equations involves finding a function that satisfies a differential equation at a specified point, often the origin. This type of problem demands both the differential equation and initial conditions to be known, laying out an exclusive trajectory for the system's evolution.

In our exercise, the initial value problem centers around the equation \( u''(x) + \lambda u(x) = 0 \) with the initial conditions \( u(0) = 0 \) and \( u'(0) = 1 \). By integrating these conditions into our equation, we direct the solution down a singular path. The solution to an initial value problem gives insights into the dynamical response of the subject system and helps in predicting future states.

Differentiating between the various scenarios of \( \lambda \) being positive, zero, or negative is crucial. Each case yields unique solutions and elucidates how a system behaves when influenced by the size and sign of such parameters, dictating the nature of the solution—be it oscillatory, linear growth, or exponential.