Question

In each exercise, (a) As in Example 3, rewrite the given scalar initial value problem as an equivalent initial value problem for a first order system. (b) Write the Euler's method algorithm, \(\mathbf{y}_{k+1}=\mathbf{y}_{k}+h\left[P\left(t_{k}\right) \mathbf{y}_{k}+\mathbf{g}\left(t_{k}\right)\right]\), in explicit form for the given problem. Specify the starting values \(t_{0}\) and \(\mathbf{y}_{0}\). (c) Using a calculator and a uniform step size of \(h=0.01\), carry out two steps of Euler's method, finding \(\mathbf{y}_{1}\) and \(\mathbf{y}_{2}\). What are the corresponding numerical approximations to the solution \(y(t)\) at times \(t=0.01\) and \(t=0.02\) ?\(y^{\prime \prime}+y=t^{3 / 2}, \quad y(0)=1, \quad y^{\prime}(0)=0\)

Short answer

Question: Convert the second order initial value problem \(y''=t^{3/2}-y\) with \(y(0)=1\) and \(y'(0)=0\) to a first order system, and carry out two steps of Euler's method with step size \(h=0.01\). Answer: The corresponding first order system is: \[\begin{cases} y_1'=y_2\\ y_2'=t^{3/2}-y_1\\ \end{cases}\] Using Euler's method, we have the following numerical approximations at times \(t=0.01\) and \(t=0.02\): \(y_1 \approx 1\) \(y_2 \approx 1.0001\)

Step-by-step solution

Convert to a first order system

Let \(y_1=y\) and \(y_2=y'\), then we have: \(y_1'=y_2\) \(y_2'=y_1''=t^{3/2}-y_1\) With the initial conditions, \(y_1(0)=1\) and \(y_2(0)=0\). Now, we have a first order system: \[\begin{cases} y_1'=y_2\\ y_2'=t^{3/2}-y_1\\ \end{cases}\] with initial values: \((t_0, \mathbf{y}_0)=(0, [1, 0]^T)\).

Write explicit Euler's method algorithm for the given problem

According to the Euler's method algorithm, we have: $$\mathbf{y}_{k+1}=\mathbf{y}_{k}+h\left[P\left(t_{k}\right) \mathbf{y}_{k}+\mathbf{g}\left(t_{k}\right)\right]$$ In our problem, we have: \(P(t) = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\), \(\mathbf{g}(t) = \begin{pmatrix} 0 \\ t^{3/2} \end{pmatrix}\), and \(\mathbf{y}_k = \begin{pmatrix} y_{1k} \\ y_{2k} \end{pmatrix}\). Therefore, the explicit Euler's method is: $ \mathbf{y}_{k+1}=\begin{pmatrix} y_{1k+1} \\ y_{2k+1} \end{pmatrix} =\begin{pmatrix} y_{1k} \\ y_{2k} \end{pmatrix} +0.01\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} y_{1k} \\ y_{2k} \end{pmatrix} +0.01\begin{pmatrix} 0 \\ t_k^{3/2} \end{pmatrix} $

Carry out two steps of Euler's method

We are given a step size \(h=0.01\). Calculate \(y_1\) and \(y_2\): \(k=0\): \(t_0=0\), \(\mathbf{y}_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\), $ \mathbf{y}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} + 0.01\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix}+ 0.01\begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ -0.01 \end{pmatrix} $ \(k=1\): \(t_1=0.01\), \(\mathbf{y}_1 = \begin{pmatrix} 1 \\ -0.01 \end{pmatrix}\), $ \mathbf{y}_2 = \begin{pmatrix} 1 \\ -0.01 \end{pmatrix} + 0.01\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ -0.01 \end{pmatrix}+ 0.01\begin{pmatrix} 0 \\ (0.01)^{3/2} \end{pmatrix} = \begin{pmatrix} 1.0001 \\ -0.019899 \end{pmatrix} $ Thus, the corresponding numerical approximations to the solution \(y(t)\) at times \(t=0.01\) and \(t=0.02\) are \(y_1 \approx 1\) and \(y_2 \approx 1.0001\), respectively.

Key concepts

Initial Value Problem

An initial value problem involves solving a differential equation with a given initial condition. It requires finding a function that solves the equation and also satisfies the initial condition specified at a particular point. For instance, in the given problem \( y'' + y = t^{3/2} \), we need to find a function \( y(t) \) that satisfies this differential equation and meets the accompanying initial conditions \( y(0) = 1 \) and \( y'(0) = 0 \).
These initial conditions provide a starting point for the solution and make it possible to tackle the problem using numerical methods like Euler's method, which step-by-step approximates the solution. Without such initial specifications, it would remain unclear which particular solution is being sought among the infinite possibilities.

First Order Systems

First order systems consist of transforming higher-order differential equations into a system of first order equations. This transformation simplifies the analysis and solution of complex equations.
In our problem, by letting \( y_1 = y \) and \( y_2 = y' \), we converted the second-order differential equation into a first order system:

• \( y_1' = y_2 \)
• \( y_2' = t^{3/2} - y_1 \)

These equations specify how \( y_1 \) and \( y_2 \) change with respect to time \( t \). The initial condition is implemented as \( y_1(0) = 1 \) and \( y_2(0) = 0 \), clearly showing the initial state of the system. This reformulation is crucial since numerical methods like Euler's method are designed to handle first-order equations seamlessly.

Numerical Approximations

Numerical approximations are methods that provide estimated solutions to differential equations, especially when an exact analytical solution is difficult to obtain. Euler's method is a straightforward technique for constructing these estimates. It works by incrementally moving from an initial value using a fixed step size, guiding the solution along with the slope provided by the differential equation.
In this problem, using a step size \( h = 0.01 \), we compute successive approximations \( \mathbf{y}_1 \) and \( \mathbf{y}_2 \). Each step slightly adjusts \( y_1 \) and \( y_2 \) using the relationship specified by the differential equation system. As a result, we obtain consecutive approximations for the solution at \( t = 0.01 \) and \( t = 0.02 \), effectively giving us a clearer picture of the solution's trajectory over time.

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They play a vital role in modeling real-world phenomena where the rate of change is involved, such as physics, engineering, and economics.
The equation given, \( y'' + y = t^{3/2} \), is a second-order differential equation. It signifies that the function \( y \) and its second derivative \( y'' \) have a combined relationship with the term \( t^{3/2} \). Solving such equations can be challenging, but techniques like converting to first-order systems and using numerical approximations provide practical ways to approximate solutions.
Through the application of numerical methods, insights into the behavior of dynamic systems as described by differential equations are possible even in scenarios where exact solutions are inaccessible.