Question

\(y^{\prime \prime}+4(1+3\) tanh \(t) y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 ; \quad 0 \leq t \leq 10\). This problem might model the motion of a spring-mass system in which the mass is released from rest with a unit initial displacement at \(t=0\) and with the spring stiffening as the motion progresses in time. Plot the numerical solutions for \(y(t)\) and \(y^{\prime}(t)\). Since tanh \(t\) approaches 1 for large values of \(t\), we might expect the solution to approximate a solution of \(y^{\prime \prime}+16 y=0\) for time \(t\) sufficiently large. Do your graphs support this conjecture?

Short answer

Answer: After obtaining the numerical solutions for \(y(t)\) and \(y^{\prime}(t)\) and plotting their graphs, analyze the behavior of the mass-spring system. If the graphs support the conjecture that the solution approximates that of \(y^{\prime\prime} + 16y = 0\) for sufficiently large values of \(t\), then the answer is yes. Otherwise, the answer is no.

Step-by-step solution

Rewrite the ODE in standard form

The given ODE is already in a standard form: $$ y^{\prime\prime} + 4(1 + 3 \text{tanh}(t))y = 0. $$

Transform the given second-order ODE into a first-order ODE

Let \(u(t) = y^{\prime}(t)\). Then, the given second-order ODE can be written as a system of two first-order ODEs as follows: $$ \begin{cases} y^{\prime}(t) = u(t) \\ u^{\prime}(t) = -4(1 + 3\text{tanh}(t))y(t). \end{cases} $$

Use a suitable numerical method to find the solution of the first-order ODE

We can use a numerical method, such as the fourth-order Runge-Kutta method (RK4), to find the solution of the system of first-order ODEs over the given interval \(0 \le t \le 10\) and with the given initial conditions \(y(0) = 1\) and \(y^{\prime}(0) = u(0) = 0\).

Plot the numerical solutions for \(y(t)\) and \(y^{\prime}(t)\)

After obtaining the numerical solutions for \(y(t)\) and \(u(t) = y^{\prime}(t)\) using the numerical method, plot both the functions with respect to \(t\) in the given interval.

Analyze the graphs and check if the conjecture is supported

Examine the graphs of \(y(t)\) and \(y^{\prime}(t)\) to see if they approach the solution of the second-order linear ODE with constant coefficients \(y^{\prime\prime} + 16y = 0\) for large values of \(t\). If the numerical plots do support this conjecture, then the behavior of the mass-spring system is consistent with what was expected.

Key concepts

Numerical Solutions

Understanding differential equations is crucial because they model a plethora of real-world phenomena, from physics to finance. However, not all differential equations have exact, analytical solutions.

When we cannot solve a differential equation analytically, numerical solutions are our best alternative. These are approximate solutions computed using computational algorithms, which give us a good idea of the behavior of the solution over a range of values. The error in these approximations can usually be made smaller by using more refined methods or increasing the computational power.

In the context of the given exercise, the second-order ODE models a spring-mass system. The exact solution is not easily obtainable because of the presence of the hyperbolic tangent function. Hence, finding a numerical solution is not just a matter of convenience; it's a necessity to understand the system's behavior.

Runge-Kutta Method

One of the most celebrated numerical methods for solving ODEs is the Runge-Kutta method, particularly the fourth-order Runge-Kutta method (RK4). The RK4 is renowned for its balance between computational efficiency and accuracy.

This method provides an iterative process to predict the behavior of the system at subsequent points, starting from known initial conditions. The 'fourth-order' part of the name indicates that the error per step is proportional to the fifth power of the step size, thus giving RK4 its high accuracy for a given step size. To use RK4 on our exercise's second-order ODE, we first convert it into a system of first-order ODEs – a necessary step because RK4 operates on first-order equations.

Second-order ODE

A second-order ODE involves derivatives of the solution function up to the second order. Our exercise deals with a second-order ODE representing a physical process —the motion of a spring-mass system.

However, most numerical methods, including RK4, are tailored for first-order ODEs. That's why we convert the original second-order ODE into a first-order system by introducing a new function for the derivative of the unknown solution. This crucial step allows us to apply the Runge-Kutta method and to find numerical solutions for the original problem, highlighting the connection between the theoretical formulation and practical numerical implementation.

Boundary Value Problems

Boundary value problems (BVPs) are types of differential equations where the solution is determined by the equation itself and specific values, or boundaries, at the ends of the domain. Unlike initial value problems, which are often solved using RK4, BVPs require different numerical methods like the shooting method or finite difference method.

The exercise presented is an initial value problem, not a boundary value problem, since it provides conditions at the initial point only, rather than at two or more boundaries. Nonetheless, understanding BVPs is essential since many physical situations, like steady-state heat distribution, are governed by this type of problem. Thus, while BVPs are not directly applied in this exercise, a firm grasp of their principles is part of a broader understanding of solving differential equations.