Question

Find the solution of the given initial value problem. Draw the graphs of the solution and of the forcing function; explain how they are related. \(y^{\mathrm{iv}}-y=u_{1}(t)-u_{2}(t) ; \quad y(0)=0, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=0, \quad y^{\prime \prime \prime}(0)=0\)

Short answer

Short Answer: The given initial value problem consists of a fourth-order linear constant-coefficient non-homogeneous differential equation and four initial conditions. The differential equation is \(y^{\mathrm{iv}}-y=u_{1}(t)-u_{2}(t)\), and the initial conditions are \(y(0)=0, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=0, \quad y^{\prime \prime \prime}(0)=0\). The general solution is given by \(y(t) = c_1e^t + c_2e^{-t} + c_3\sin(t) + c_4\cos(t) + A(t)[u_1(t)-u_2(t)]\), where \(c_1, c_2, c_3,\) and \(c_4\) are determined by initial conditions, and \(A(t)\) is determined by integrating the forcing function. The forcing function directly affects the particular solution, while the homogeneous solution remains independent. The specific form of the solution, forcing function, and their graphs cannot be determined without explicit expressions for \(u_1(t)\) and \(u_2(t)\).

Step-by-step solution

Identify the differential equation and initial conditions

The given IVP consists of a fourth-order linear constant-coefficient non-homogeneous differential equation (DE) and four initial conditions: DE: \(y^{\mathrm{iv}}-y=u_{1}(t)-u_{2}(t)\) Initial conditions: \(y(0)=0, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=0, \quad y^{\prime \prime \prime}(0)=0\)

Solve the homogeneous part of the differential equation

First, we need to find the general solution for the homogeneous part of the DE, which is \(y^{\mathrm{iv}}-y=0\). To do that, we assume a solution of the form \(y(t)=e^{rt}\), where r is a constant: \((r^4 - 1)e^{rt} = 0\) The above equation is true only if \(r^4 = 1\), which gives the eigenvalues \(r = \pm 1, \pm i\). Then, the homogeneous solution is: \(y_h(t) = c_1e^t + c_2e^{-t} + c_3\sin(t) + c_4\cos(t)\)

Find the particular solution for the non-homogeneous part

For this step, we need to find the particular solution \(y_p(t)\) of the following DE: \(y^{\mathrm{iv}}-y=u_{1}(t)-u_{2}(t)\). To obtain it, we will first suppose that \(y_p(t)=A[u_1(t)-u_2(t)]\), then differentiate it four times and substitute it into the DE. Integrate each term in sequence to find the unknown function \(A(t)\), and thus determine the particular solution. Since the forcing function is not given explicitly, let's simply denote it as: \(y_p(t) = A(t)[u_1(t)-u_2(t)]\)

Combine homogeneous and particular solutions to find the general solution

Now, we need to find the general solution of the given DE. We can find it by adding the homogeneous solution and the particular solution that we found previously: \(y(t) = y_h(t) + y_p(t) = c_1e^t + c_2e^{-t} + c_3\sin(t) + c_4\cos(t) + A(t)[u_1(t)-u_2(t)]\)

Apply initial conditions to find the particular solution

Next, we will apply the initial conditions to the general solution above in order to find the particular solution that satisfies them. By applying each initial condition, we will have a system of equations that will determine the values of the constants \(c_1, c_2, c_3,\) and \(c_4\). Since the forcing function is not given explicitly, this step cannot be completed fully. However, the particular solution would be in the form: \(y(t) = c_1e^t + c_2e^{-t} + c_3\sin(t) + c_4\cos(t) + A(t)[u_1(t)-u_2(t)]\), where \(c_1, c_2, c_3,\) and \(c_4\) are determined by the initial conditions.

Draw the graphs of the solution and forcing function

To draw the graphs, we need the forms of the particular solution and the forcing function \(u_1(t)-u_2(t)\). Because the forcing function is not given explicitly, we cannot draw their graphs directly. However, we can analyze the relationship between the solution and the forcing function by looking at the particular solution we found.

Explain the relationship between the solution and the forcing function

The relationship between the solution and the forcing function can be seen in the particular solution \(y_p(t) = A(t)[u_1(t)-u_2(t)]\). The forcing function determines the particular solution of the given DE, as the function A(t) is determined by integrating the forcing function. The homogeneous solution represents the free response of the DE and is independent of the forcing function. In conclusion, the forcing function directly affects the form of the particular solution, while the homogeneous solution remains independent.

Key concepts

Fourth-order differential equations

In differential equations, the order is determined by the highest derivative present. A fourth-order differential equation includes the fourth derivative of the unknown function. Such equations are common in modeling real-world phenomena, including physics and engineering.
In more specific terms, the general form of a fourth-order differential equation can be expressed as: \[ a_4 y^{(iv)} + a_3 y^{(iii)} + a_2 y^{\prime\prime} + a_1 y^{\prime} + a_0 y = g(t) \]
Here,

• \(y^{(iv)}\) is the fourth derivative of \(y\).
• \(a_4, a_3, a_2, a_1, \) and \(a_0\) are constant coefficients.
• \(g(t)\) is a given function that could depend on \(t\).

The challenge with fourth-order equations often lies in solving them effectively, needing a broader set of methods and approaches than lower-order equations. Understanding them is crucial for comprehensive studies in mathematical modeling.

Non-homogeneous differential equations

Non-homogeneous differential equations are equations with a term that is not equal to zero, unlike homogeneous equations that equal zero. This non-zero term often represents some external force acting on the system, hence the term 'forcing function'.
The general structure of non-homogeneous differential equations is: \[ L(y) = f(t) \] Here,

• \(L(y)\) is a linear differential operator applied to \(y\).
• \(f(t)\) is the non-zero function which makes the equation non-homogeneous.

These equations are crucial in modeling real-world systems because they often account for inputs or external influences, making them more applicable to practical scenarios than their homogeneous counterparts. Solving these requires finding both the homogeneous and particular solutions, which together form the general solution.

Homogeneous solution

The homogeneous solution is a part of the solution to a differential equation where the equation is set to zero: \(L(y) = 0\). It captures the behavior of the system when no external force is applied.
In the context of a fourth-order differential equation, this involves solving the related characteristic equation. The solutions to this characteristic equation give us the necessary constants (or functions like sine and cosine) to detail the homogeneous solution. For instance:

• A characteristic equation like \(r^4 - 1 = 0\) might suggest solutions like \(r = \pm 1, \pm i\).
• The homogeneous solution based on these might be \(y_h(t) = c_1 e^t + c_2 e^{-t} + c_3 \sin(t) + c_4 \cos(t)\).

The homogeneous solution is essential for understanding the inherent response of the system without any external influences.

Forcing function in differential equations

The forcing function in a differential equation is the term that makes the differential equation non-homogeneous. It represents an input or a driving force acting on the system, which affects the particular part of the solution.
In our example, the forcing function is represented as \(u_1(t) - u_2(t)\). This part of the equation compels the system to respond by seeking a particular solution that meets the equation's overall conditions with this external influence in mind.
Forcing functions have practical applications, such as modeling a mechanical system's response to an external force. For instance, in electrical circuits, a forcing function could mimic a voltage or current applied to the system.

• The term influences the particular solution \(y_p(t)\), acting as the system's driven response.
• The specific nature of the forcing function will guide how the particular solution is derived and influences the overall general solution of the differential equation.

Understanding the forcing function is crucial because it defines how external factors will modify the system's behavior.