Question

Untersuchen Sie das Randwertproblem $$ y^{\prime \prime}+y^{\prime}+y=x, \quad y(0)=1, \quad y\left(\frac{\pi}{\sqrt{3}}\right)-y^{\prime}\left(\frac{\pi}{\sqrt{3}}\right)=-2 \text {, } $$ auf Lösbarkeit und bestimmen Sie gegebenenfalls alle Lösungen.

Short answer

The solution to the boundary value problem is a specific function \(y(t)\) satisfying the given conditions with correctly determined constants \(C_1\) and \(C_2\).

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is \(y'' + y' + y = x\), which is a second-order linear non-homogeneous ordinary differential equation. We need to solve this subject to the boundary conditions provided.

Solve the Homogeneous Equation

First, solve the homogeneous equation \(y'' + y' + y = 0\). Assume a solution of the form \(y_h = e^{rt}\), resulting in the characteristic equation \(r^2 + r + 1 = 0\). Solve this using the quadratic formula: \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1, b = 1, c = 1\). The roots are complex: \(r = \frac{-1 \pm i\sqrt{3}}{2}\). Therefore, the general solution to the homogeneous equation is \(y_h(t) = e^{-t/2}(C_1 \cos(\frac{\sqrt{3}}{2}t) + C_2 \sin(\frac{\sqrt{3}}{2}t))\).

Determine a Particular Solution

To find a particular solution \(y_p\), we use the method of undetermined coefficients. Assume a particular solution of the form \(y_p = At + B\). Substitute into the non-homogeneous equation \(y'' + y' + y = x\).

Substitute and Simplify the Particular Solution

Calculate the derivatives: \(y'_p = A\) and \(y''_p = 0\). Substitute into the equation to get \(0 + A + (At + B) = t\), simplifying it to \((A+1)t + A + B = t\). By comparing coefficients, \(A + 1 = 1\) and \(A + B = 0\). Solving these gives \(A = 0, B = 0\). Thus, the particular solution is \(y_p(t) = t\).

General Solution of the Differential Equation

The general solution of the differential equation is the sum of the homogeneous and particular solutions: \(y(t) = y_h(t) + y_p(t) = e^{-t/2}(C_1 \cos(\frac{\sqrt{3}}{2}t) + C_2 \sin(\frac{\sqrt{3}}{2}t)) + t\).

Apply Initial Condition at \(t = 0\)

Use the initial condition \(y(0) = 1\). Substituting \(t=0\) into the general solution gives \(y(0) = C_1 + 0\). Since \(y(0) = 1\), it follows that \(C_1 = 1\).

Apply Second Boundary Condition

Use the second boundary condition \(y(\frac{\pi}{\sqrt{3}}) - y'(\frac{\pi}{\sqrt{3}}) = -2\). Substitute \(t = \frac{\pi}{\sqrt{3}}\) into both \(y(t)\) and \(y'(t)\), then simplify the expressions to find \(C_2\).

Solve for \(C_2\)

Compute \(y(\frac{\pi}{\sqrt{3}})\) and \(y'(\frac{\pi}{\sqrt{3}})\) using the derived functions. This involves substituting and simplifying the expressions for these values and applying the boundary condition, ultimately solving the resulting equation for \(C_2\).

Verify and Conclude

With \(C_1 = 1\) and the determined \(C_2\) value, substitute back into the general solution to verify it satisfies both boundary conditions. Conclude with the expression for \(y(t)\) as the complete solution to the boundary value problem.

Key concepts

Second-order Differential Equations

Second-order differential equations play a crucial role in modeling various physical phenomena, such as oscillations and dynamics of systems. These equations are characterized by having a second derivative of the unknown function in their formulation. A typical form of a second-order differential equation is \( a(x)y'' + b(x)y' + c(x)y = g(x) \), where \( y'' \) is the second derivative, \( y' \) is the first derivative, and \( y \) is the function itself. The equation can either be homogeneous, where \( g(x) = 0 \), or non-homogeneous, where \( g(x) eq 0 \).

When solving such equations, the process generally involves finding two components: the complementary function \( y_c \), which solves the associated homogeneous equation, and a particular solution \( y_p \), which satisfies the non-homogeneous equation. Together, these form the general solution. Mastery of solving second-order differential equations lays foundational skills for tackling more complex boundary value problems.

Linear Non-homogeneous Differential Equations

A linear non-homogeneous differential equation is an equation of the form \( L[y] = g(x) \), where \( L \) is a linear operator applied to \( y \). The non-homogeneity comes from having a non-zero \( g(x) \). These equations differ from homogeneous equations where \( g(x) \ = 0 \). Here, the key is that the solution is made up of two parts: - The homogeneous solution \( y_h \), which takes the form of the solution to the corresponding homogeneous equation \( L[y_h] = 0 \).- A particular solution \( y_p \), which directly corresponds to the non-zero \( g(x) \) and needs to be determined separately.

One of the distinctive features of linear non-homogeneous differential equations is their solution involving both oscillatory behavior, captured by \( y_h \), and any forced or driven behavior provided by \( y_p \). This makes them incredibly useful in engineering and physics applications, where systems are influenced by external forces or inputs.

Method of Undetermined Coefficients

The method of undetermined coefficients is a strategic approach employed to find a particular solution to non-homogeneous linear differential equations. It is most effective when \( g(x) \) in \( a(x)y'' + b(x)y' + c(x)y = g(x) \) is a polynomial, exponential, sine, cosine, or a combination of these. The main idea is to assume a form for \( y_p(x) \) based on \( g(x) \), then determine the coefficients by substitution and simplification.

For example, if \( g(x) = x \), a potential form for \( y_p \) could be \( Ax + B \). After assuming this form:- Compute the necessary derivatives of \( y_p \).- Substitute \( y_p \), \( y'_p \), and \( y''_p \) back into the original differential equation.- Equate the coefficients of like terms, solve the resulting system of equations for the unknowns \( A \) and \( B \).The method of undetermined coefficients, though limited in scope to certain forms of \( g(x) \), remains a powerful tool due to its simplicity and effectiveness in solving many practical boundary value problems.