Question

In each exercise, convert the given problem into a corresponding boundary value problem for the Prufer variables \(R\) and \(\theta\); see equation (18). State both the differential equations and the boundary conditions. Assume \(R(a)\) and \(R(b)\) are nonzero. $$ \begin{aligned} &u^{\prime \prime}-2 u^{\prime}=-\lambda t u, \quad 0

Short answer

Given the boundary value problem: $$u'' - 2u' = -\lambda t u, \quad 0 < x < 2$$ with boundary conditions: $$u(0) = 0, \quad u(2) = 0,$$ find the boundary value problem for the Prufer variables R and θ, given that R(a) and R(b) are nonzero. The boundary value problem for the Prufer variables R and θ is given by: $$ \begin{cases} R''(x) &= \lambda t R(x) \\ R(x)\theta''(x) &= R'(x)\theta'(x) \\ \cos{\theta(0)} &= 0 \\ \cos{\theta(2)} &= 0 \end{cases} $$

Step-by-step solution

Introduce Prufer variable substitution

To convert the given differential equation into the Prufer form, let's use the substitution: $$u(x) = R(x)\cos{\theta(x)}$$ Now differentiate both sides of this equation with respect to \(x\). Step 2: Differentiate the substitution

Differentiate the Prufer substitution

Differentiating \(u(x) = R(x)\cos{\theta(x)}\) with respect to \(x\) gives: $$u'(x) = R'(x)\cos{\theta(x)} - R(x)\sin{\theta(x)}\theta'(x)$$ Now differentiate \(u'(x)\) with respect to \(x\): $$u''(x) = R''(x)\cos{\theta(x)} - 2R'(x)\sin{\theta(x)}\theta'(x) - R(x)\cos{\theta(x)}\theta'^2(x) - R(x)\sin{\theta(x)}\theta''(x)$$ Step 3: Substitute the expressions into the original equation

Substitute and simplify the equation

Substituting the expressions of \(u(x)\), \(u'(x)\), and \(u''(x)\) into the differential equation and equating the coefficients of \(\cos\theta(x)\) and \(\sin\theta(x)\), we obtain the following system of equations: $$ \begin{cases} R''(x) &= \lambda t R(x) \\ R'(x)\theta'(x) + R(x)\theta''(x) &= 2R'(x)\theta'(x) \end{cases} $$ Step 4: Solve for the differential equations for R and θ

Simplify the differential equations

Solving these equations, we get: $$ \begin{cases} R''(x) &= \lambda t R(x) \\ R(x)\theta''(x) &= R'(x)\theta'(x) \end{cases} $$ Now, we have the differential equations for R and θ. Step 5: Find the boundary conditions for R and θ

Apply the given boundary conditions

Applying the given boundary conditions on the expression \(u(x) = R(x)\cos{\theta(x)}\), we get: $$R(0)\cos{\theta(0)} = 0$$ Since \(R(0)\) is nonzero, we must have: $$\cos{\theta(0)} = 0$$ Similarly, $$R(2)\cos{\theta(2)} = 0$$ Since \(R(2)\) is nonzero, we have: $$\cos{\theta(2)} = 0$$ Step 6: Finalize the boundary value problem for R and θ

Write down the final boundary value problem

In summary, we have the following boundary value problem for the Prufer variables R and θ: $$ \begin{cases} R''(x) &= \lambda t R(x) \\ R(x)\theta''(x) &= R'(x)\theta'(x) \\ \cos{\theta(0)} &= 0 \\ \cos{\theta(2)} &= 0 \end{cases} $$

Key concepts

Differential Equations

Differential equations are mathematical tools that describe relationships involving functions and their derivatives. They are peculiar because, rather than providing a direct solution, they set a condition under which a function can be determined.

In our exercise, the differential equation given is a second-order linear ordinary differential equation (ODE), meaning it involves the second derivative of a function, namely, \(u(x)\). The specific form \(u''(x) - 2u'(x) = -\text{λ}t u(x)\) characterizes how the unknown function, in this case \(u\), and its derivatives relate to one another along with varying positions \(x\) and involves an eigenvalue \(\text{λ}\). The solution to this differential equation is not apparent. The strategy involves transforming the original problem using Prufer variable substitution to simplify the solution process.

Boundary Conditions

Boundary conditions are constraints necessary for the unique determination of a solution to a differential equation. These conditions are provided at the boundaries of the domain where the solution is sought. In our exercise, the boundary conditions are \(u(0) = 0\) and \(u(2) = 0\), which impose that the function \(u(x)\) should vanish at both ends of the interval \([0,2]\).

When these conditions are applied to the transformed equations using Prufer variables, we derive that \(R(0)\cos{\theta(0)} = 0\) and \(R(2)\cos{\theta(2)} = 0\). Given that \(R(0)\) and \(R(2)\) are non-zero, the only way to satisfy these conditions is if the cosine of the variables \(\theta(0)\) and \(\theta(2)\) are zero. These provide us with significant information about the possible values of \(\theta(x)\) at the boundaries, aiding in the search for a solution to the boundary value problem.

Prufer Variable Substitution

Prufer variable substitution is a technique used to simplify certain types of differential equations into a form that's more manageable. The substitution introduces two new variables, \(R\) and \(\theta\), representing the amplitude and phase functions. Specifically, the substitution is \(u(x) = R(x)\cos{\theta(x)}\).

The magic in this substitution lies in transforming the problem from one involving the function \(u(x)\) and its derivatives to one that represents the same situation through the behavior of \(R(x)\) and \(\theta(x)\). Differentiating and substituting back into the original equation, the problem becomes much easier to interpret and solve, breaking down the complex, original ODE into a set of potentially more straightforward differential equations.

Eigenvalue Problems

Eigenvalue problems are a specific category of problems in which a scalar \(\text{λ}\) alters a vector without changing its direction in a linear transformation described by an operator. In differential equations, an eigenvalue is a special number that, when used as a parameter in the equation, allows for non-trivial solutions.

In our differential equation context, the term \(-\text{λ}t u\) suggests that we are dealing with an eigenvalue problem where \(\text{λ}\) affects the shape and properties of the solution function. The solution is not just any function that satisfies the differential equation but one that also adheres to the boundary conditions. These problems often arise in physics and engineering, dealing with quantities such as vibrations and modes of structures, emphasizing their practical relevance and the need for familiarization by students.