Question

In Exercises 26-30 find necessary and sufficient conditions on \(\alpha, \beta, \rho,\) and \(\delta\) for the boundary value problem to have a unique solution for every continuous \(F,\) and find the Green's function. $$ y^{\prime \prime}+y=F(x), \quad \alpha y(0)+\beta y^{\prime}(0)=0, \quad \rho y(\pi / 2)+\delta y^{\prime}(\pi / 2)=0 $$

Short answer

In summary, the necessary and sufficient condition for the given boundary value problem to have a unique solution for every continuous function \(F(x)\) is \(\alpha\rho+\beta\delta\neq 0\). The Green's function is given by \(G(x,x_0) = A(x)\cos(x_0)+B(x)\sin(x_0)\), where \(A(x)\) and \(B(x)\) are derived from the integrals involving the delta function.

Step-by-step solution

Analyze the homogeneous differential equation

Consider the homogeneous differential equation associated with the given problem: $$ y^{\prime \prime}+y=0. $$ This equation has the general solution given by: $$ y(x)=C_1\cos(x)+C_2\sin(x), $$ where \(C_1\) and \(C_2\) are arbitrary constants. Now, we want to analyze how these constants are related to the given boundary conditions.

Apply the boundary conditions

Apply the first boundary condition: $$ \alpha y(0)+\beta y^{\prime}(0)=0. $$ Substitute the general solution and its derivative into this boundary condition to get: $$ \alpha\left(C_1\cos(0)+C_2\sin(0)\right)+\beta\left(-C_1\sin(0)+C_2\cos(0)\right)=\alpha C_1 + \beta C_2 = 0. $$ Next, apply the second boundary condition: $$ \rho y\left(\frac{\pi}{2}\right)+\delta y^{\prime}\left(\frac{\pi}{2}\right)=0. $$ Substitute the general solution and its derivative to get: $$ \rho\left(C_1\cos\left(\frac{\pi}{2}\right)+C_2\sin\left(\frac{\pi}{2}\right)\right)+\delta\left(-C_1\sin\left(\frac{\pi}{2}\right)+C_2\cos\left(\frac{\pi}{2}\right)\right)=\rho C_2 - \delta C_1 = 0. $$

Determine the necessary and sufficient conditions

We now have a pair of equations for \(C_1\) and \(C_2\): $$ \alpha C_1 + \beta C_2=0 \\ \rho C_2 - \delta C_1 = 0. $$ For a unique solution to the inhomogeneous problem to exist for every continuous function \(F(x)\), the homogeneous problem must have only the trivial solution, \(y(x)=0\). This means that the determinant of the coefficients of the above system of equations must be nonzero: $$ \begin{vmatrix} \alpha & \beta\\ -\delta & \rho \end{vmatrix}=\alpha\rho+\beta\delta\neq0. $$ This is the necessary and sufficient condition on \(\alpha, \beta, \rho,\) and \(\delta\) for the boundary value problem to have a unique solution for every continuous \(F(x)\).

Find the Green's function

To find the Green's function, let us consider the inhomogeneous equation $$ y^{\prime \prime}+y=\delta(x-x_0). $$ The general solution of this equation is a combination of the general solution of the homogeneous equation and a particular solution of the inhomogeneous equation: $$ y(x)=C_1\cos(x)+C_2\sin(x)+y_p(x). $$ The particular solution \(y_p(x)\) must satisfy the given boundary conditions. Using the method of variation of parameters, we find that $$ y_p(x) = A(x)\cos(x_0)+B(x)\sin(x_0), $$ where $$ A(x)=\int\sin(x)\delta(x-x_0)dx $$ and $$ B(x)=\int-\cos(x)\delta(x-x_0)dx. $$ Thus, the Green's function, which represents the response of the system to an impulse at location \(x_0\), is given by $$ G(x,x_0) = A(x)\cos(x_0)+B(x)\sin(x_0). $$

Key concepts

Differential Equations

Understanding differential equations is essential for solving a plethora of problems in physics, engineering, and other fields. These equations inherently describe how one variable changes with respect to another. Specifically, when dealing with an equation like
\( y^{\textprime \textprime} + y = F(x) \), we encounter a second-order linear ordinary differential equation, where \( y^{\textprime \textprime} \) represents the second derivative of \( y \) with respect to \( x \), and \( F(x) \) is a given function. The general solution of this homogeneous equation is \( y(x) = C_1\cos(x) + C_2\sin(x) \), where \( C_1 \) and \( C_2 \) are constants that will be determined by the boundary conditions of the problem. It's crucial to grasp that this combination of sine and cosine functions encompasses all possible solutions to this type of homogeneous differential equation.

Unique Solution Conditions

While differential equations can have many solutions, certain problems require a single, unique solution. This uniqueness is particularly important in boundary value problems where we use conditions at distinct points to pinpoint the exact solution we're looking for.
For instance, given the boundary conditions \( \alpha y(0) + \beta y^{\textprime}(0) = 0 \) and \( \rho y(\frac{\pi}{2}) + \delta y^{\textprime}(\frac{\pi}{2}) = 0 \), the unique solution to our differential equation exists only if the related homogeneous system has the trivial solution \( y(x)=0 \). This boils down to ensuring that the determinant of the coefficients matrix is nonzero:\(\begin{vmatrix}\alpha & \beta\-\delta & \rho\end{vmatrix} = \alpha\rho + \beta\delta eq 0\).This criterion guarantees that there is no nontrivial solution that fits the boundary conditions, thus enforcing the uniqueness of the solution when we include a non-homogeneous term \( F(x) \).

Green's Function

Green's function is a powerful tool to analyze inhomogeneous differential equations with boundary conditions. It essentially represents the influence that a point within the domain has on the values throughout the zone of interest. To obtain Green's Function, one typically starts with an impulse function, like \( \delta(x - x_0) \), and works out the system's response to it.
In our exercise, we find a particular solution by including the impulse function:
\( y_p(x) = A(x)\cos(x_0) + B(x)\sin(x_0) \),where \( A(x) \) and \( B(x) \) are obtained using the method of variation of parameters with integrals involving the delta function. The Green's function, \( G(x,x_0) \), encapsulates how a perturbation at \( x_0 \) impacts the system. Understanding and determining Green's function is not only fundamental for this problem but also for a broad range of physical systems described by differential equations.