Question

In Exercises 12-15 find a formula for the solution of the boundary problem by the method used to prove Theorem \(13.1 .3 .\) Assume that \(a

Short answer

To find the solution to the boundary value problem with the given ODE \(y^{\prime \prime} - y = F(x)\) and boundary conditions \(y(a) = 0\) and \(y^{\prime}(b) = 0\), we used the method of Green's function. We first found two solutions to the homogeneous ODE \(u_1(x) = e^x\) and \(u_2(x) = e^{-x}\), then found the functions \(U_1(x) = 1 - e^x\) and \(U_2(x) = e^{x} - e^{2x}\) that satisfy the boundary conditions, and constructed the Green's function \(G(x, t)\) using these functions. The general solution to the boundary value problem can be expressed as an integral involving the Green's function and the given function \(F(x)\): $$ y(x) = \int_{a}^{b} F(t) \left[ (1 - e^x)(e^t - e^{2t}) \right] dt + \int_{a}^{b} F(t) \left[ (1 - e^t)(e^x - e^{2x}) \right] dt $$

Step-by-step solution

Define the ODE and boundary conditions

We are given the following second-order ODE with boundary conditions: $$ y^{\prime \prime} - y = F(x), \quad y(a) = 0, \quad y^{\prime}(b) = 0 $$

Find the Green's function

In order to find the Green's function, we first need to find two solutions, \(u_1(x)\) and \(u_2(x)\), to the homogeneous ODE: $$ y^{\prime \prime} - y = 0 $$ The characteristic equation for this ODE is: $$ \lambda^2 - 1 = 0 $$ Solving this equation gives us \(\lambda = \pm 1\). Therefore, the two linearly independent solutions are \(u_1(x) = e^x\) and \(u_2(x) = e^{-x}\). Now we need to find two functions, \(U_1(x)\) and \(U_2(x)\), such that: $$ U_1^{\prime \prime} - U_1 = 0, \quad U_1(a) = 0, \quad U_1^{\prime}(a) = u_1^{\prime}(a) $$ and $$ U_2^{\prime \prime} - U_2 = 0, \quad U_2(b) = 0, \quad U_2^{\prime}(b) = u_2^{\prime}(b) $$ Since \(u_1(x) = e^x\) and \(u_2(x) = e^{-x}\), we have \(u_1^{\prime}(x) = e^x\) and \(u_2^{\prime}(x) = -e^{-x}\). Therefore: $$ U_1^{\prime}(a) = e^a, \qquad U_2^{\prime}(b) = -e^{-b} $$ We can find the functions \(U_1(x)\) and \(U_2(x)\) by using the method of variation of parameters: $$ U_1(x) = u_1(x) \int \frac{u_2(x) W(x)}{u_1(x) u_2^{\prime}(x) - u_1^{\prime}(x) u_2(x)} dx, \qquad U_2(x) = u_2(x) \int \frac{u_1(x) W(x)}{u_1(x) u_2^{\prime}(x) - u_1^{\prime}(x) u_2(x)} dx $$ where \(W(x) = u_1(x) u_2^{\prime}(x) - u_1^{\prime}(x) u_2(x)\) is the Wronskian of \(u_1(x)\) and \(u_2(x)\). Calculating the Wronskian, we get: $$ W(x) = e^x(-e^{-x}) - e^x(e^{-x}) = -2 $$ Thus, we have: $$ U_1(x) = e^x \int \frac{e^{-x}(-2)}{-2} dx, \qquad U_2(x) = e^{-x} \int \frac{e^x(-2)}{-2} dx $$ Integrating each function, we obtain: $$ U_1(x) = e^x \int e^{-x} dx, \qquad U_2(x) = e^{-x} \int e^x dx $$ $$ U_1(x) = e^x(-e^{-x}), \qquad U_2(x) = e^{-x}(e^{x}) $$ Therefore, we have found the functions: $$ U_1(x) = 1 - e^x, \qquad U_2(x) = e^{x} - e^{2x} $$ Now, we can find the Green's function \(G(x, t)\): $$ G(x, t) = \left \{ \begin{array}{ll} U_1(x) U_2(t) & \quad x \leq t \\ U_1(t) U_2(x) & \quad t \leq x \end{array} \right. $$ So, the Green's function is: $$ G(x, t) = \left\{ \begin{array}{ll} (1 - e^x)(e^t - e^{2t}) & \quad x \leq t \\ (1 - e^t)(e^x - e^{2x}) & \quad t \leq x \end{array} \right. $$

Solve the ODE using the Green's function

Now that we have the Green's function, we can find the solution to the given ODE by integrating it with the function \(F(x)\): $$ y(x) = \int_{a}^{b} F(t) G(x, t) dt $$ Without knowing the specific function given by \(F(x)\), we cannot calculate the final solution. However, this formula provides a general solution for any given \(F(x)\): $$ y(x) = \int_{a}^{b} F(t) \left[ (1 - e^x)(e^t - e^{2t}) \right] dt + \int_{a}^{b} F(t) \left[ (1 - e^t)(e^x - e^{2x}) \right] dt $$

Key concepts

Green's Function Method

The Green's function method is a powerful technique for solving boundary value problems involving linear differential equations. The essence of this method is to construct a Green's function, which serves as the kernel of an integral operator transforming the source term of the differential equation into its solution. In context of the exercise, the goal was to solve the boundary value problem for the second-order linear ordinary differential equation (ODE):
\[\begin{equation}y^{\'\'} - y = F(x), \quad y(a) = 0, \quad y^\'(b) = 0\end{equation}\]
by finding the Green's function. This was achieved by first solving a homogeneous ODE to find two linearly independent solutions and then leveraging these to construct the Green's function. The final step was to integrate the Green's function with the non-homogeneous term, F(x), giving us a formula for the general solution of the ODE.

• The Green's function encapsulates all the information about the boundary conditions and the structure of the ODE.
• The method can be extended to multiple dimensions and more complex boundary conditions.
• For different problems, finding the Green's function involves adapting the construction process to fit the specific boundary conditions.

Second-order Ordinary Differential Equations

Second-order ordinary differential equations are equations that involve the unknown function y, its first derivative y', and its second derivative y''. These are crucial for describing numerous physical phenomena, such as motions governed by Newton's law of motion. In the exercise, the second-order ODE:
\[\begin{equation}y^{\'\'} - y = F(x)\end{equation}\]
was provided with specific boundary conditions to be solved.

• These ODEs typically require two initial or boundary conditions to find a unique solution, as two initial values are needed to integrate twice.
• Finding analytical solutions might not always be possible; hence, numerical methods or methods like Green's function become valuable.
• The characteristic equation aids in finding homogeneous solutions, which play a critical role in constructing Green's functions for inhomogeneous equations.

By solving the characteristic equation, we get particular solutions to the homogeneous equation, which are used as building blocks for the more complicated solutions of inhomogeneous equations.

Homogeneous ODE Solutions

Homogeneous solutions to second-order ODEs are found by setting the non-homogeneous term (typically denoted as F(x) in an inhomogeneous equation) to zero. The resulting homogeneous equation describes the complementary or natural behavior of the system without external forces. In the exercise, the homogeneous ODE:
\[\begin{equation}y^{\'\'} - y = 0\end{equation}\]
was addressed, which has the general solution based on a characteristic equation.

Characteristic Equation

For the exercise under consideration, the characteristic equation is \[\begin{equation}\lambda^2 - 1 = 0\end{equation}\]which yields the eigenvalues \(\lambda = -1\) and \(\lambda = 1\). These eigenvalues lead to the homogeneous solutions \(e^x\) and \(e^{-x}\), respectively.

• Homogeneous solutions are crucial for constructing the Green's function in a boundary value problem.
• They form the complementary part of the general solution to the entire inhomogeneous problem.
• Understanding these solutions helps us grasp the underlying structure of the differential equation before any external forcing is introduced.

Variation of Parameters

Variation of parameters is a method applied to non-homogeneous differential equations, where the solution comprises a homogeneous part and a particular part due to the non-homogeneous term. This technique adjusts the constants in the complementary solution into functions that accommodate the specific structure of the non-homogeneous term, F(x), in this case. The exercise demonstrates this method to find the functions \(U_1(x)\) and \(U_2(x)\) that meet the required conditions at points a and b for the defined ODE.

Application in the Exercise

The variation of parameters strategy was used to modify the solutions \(u_1(x)\) and \(u_2(x)\) to fulfill the boundary conditions, leading to formulas for \(U_1(x)\) and \(U_2(x)\).

• This method is a sophisticated technique that requires knowledge of the solutions to the associated homogeneous problem.
• It is particularly useful when the inhomogeneous term, F(x), does not permit easy finding of a particular solution.
• The Wronskian, a determinant related to the homogeneous solutions, is a key part of the computation.

Overall, variation of parameters complements the Green's function method in solving boundary value problems for second-order ODEs in a structured manner.