Question

For the given initial value problem, (a) Execute 20 steps of the Taylor series method of order \(p\) for \(p=1,2,3\). Use step size \(h=0.05 .\) (b) In each exercise, the exact solution is given. List the errors of the Taylor series method calculations at \(t=1\). \(y^{\prime}=\frac{1+y^{2}}{1+t}, \quad y(0)=0 . \quad\) The exact solution is \(y(t)=\tan [\ln (1+t)]\)

Short answer

Question: Calculate the errors for p = 1, 2, and 3 at t = 1 for the Taylor series method with a step size h = 0.05. The initial value problem is \(y'(t) = \frac{1 + y(t)^2}{1 + t}\) and \(y(0) = 0\). The exact solution is given as \(y(t) = \tan[\ln(1+t)]\).

Step-by-step solution

Determine the general Taylor series formula for p = 1, 2, and 3

The Taylor series formula is given by: \(y(t+h) = y(t) + h\cdot y'(t) + \frac{h^2}{2!}\cdot y''(t) + ... + \frac{h^p}{p!} \cdot y^{(p)}(t) + R_p(t)\) For p = 1, 2, and 3: - \(p = 1\): \(y(t+h) = y(t) + h\cdot y'(t)\) - \(p = 2\): \(y(t+h) = y(t) + h\cdot y'(t) + \frac{h^2}{2!}\cdot y''(t)\) - \(p = 3\): \(y(t+h) = y(t) + h\cdot y'(t) + \frac{h^2}{2!}\cdot y''(t) + \frac{h^3}{3!}\cdot y^{(3)}(t)\)

Find the first, second, and third derivatives of y(t)

To find the first, second, and third derivatives \((y'(t), y''(t), y^{(3)}(t))\), differentiate the given equation \(y'(t) = \frac{1+y(t)^2}{1+t}\) with respect to t: - \(y'(t) = \frac{1+y(t)^2}{1+t}\) Differentiating once with respect to t: - \(y''(t) = -(1+y^2)^2\frac{y'}{(1+t)^3}+2\frac{yy'}{(1+t)^2}\) Differentiating twice with respect to t: - \(y^{(3)}(t) = -6\frac{y'(\frac{y^2}{(1+t)^5}+1+(\frac{y^2}{(1+t)^3})^2)+\frac{y'}{(1+t)^3}}{y'}+\frac{4y'}{(1+t)^3}+\frac{2y(\frac{y'}{(1+t)^3}-\frac{3y'}{(1+t)^4})}{1+t}\)

Execute 20 steps of the Taylor series method for p = 1, 2, and 3

To find the approximate solutions at \(t=1\) with a step size of \(h=0.05\), use the p = 1, 2, and 3 Taylor series formulas with the corresponding derivatives. You'll need to start with \(y(0)=0\) and calculate each step iteratively. Note that in these calculations, y'(t) will be replaced with the given initial value problem equation.

Calculate the exact solution at t = 1

The exact solution is given by: \(y(t) = \tan[\ln(1+t)]\) Plug t = 1 into the exact solution equation: \(y(1) = \tan[\ln(1+1)] = \tan[\ln(2)]\)

Calculate errors for p = 1, 2, and 3 at t = 1

To calculate the errors for p = 1, 2, and 3 at t = 1, compare each Taylor series approximation with the exact solution from Step 4: - Error for p = 1: \(Error = |\tan[\ln(2)] - T_1(1)|\) - Error for p = 2: \(Error = |\tan[\ln(2)] - T_2(1)|\) - Error for p = 3: \(Error = |\tan[\ln(2)] - T_3(1)|\) Given the complexity of the Taylor series approximations, it is recommended to use a numerical computation tool to perform Steps 3 and 5.

Key concepts

Initial Value Problems

Initial value problems are a fundamental part of differential equations where the solution is determined by an initial set of conditions. These problems are crucial in many real-world applications, such as physics, engineering, and biology. The typical format is a differential equation paired with an initial condition, where the solution must satisfy both. For the exercise provided, the initial condition is noted as \(y(0) = 0\), with the differential equation being \(y' = \frac{1+y^2}{1+t}\).

This means we want to find a function \(y(t)\) that not only results in the given derivative when plugged into the left side of the equation but also begins with the value \(y=0\) at \(t=0\). Initial value problems are often solved using numerical methods when analytical solutions are challenging or impossible to find.

Numerical Methods

Numerical methods are powerful tools used to approximate solutions to mathematical problems that can't be solved analytically, especially with differential equations. In this exercise, a Taylor series method is employed, which is a specific kind of numerical method. This method expands a function into an infinite sum of terms calculated from the values of its derivatives at a single point.

With the Taylor series, since we're working initially with \(p = 1, 2,\) and \(3\), this corresponds to using the first, second, and third derivatives, respectively. It allows us to execute 20 steps to approximate the solution of the initial value problem at discrete points. This stepwise approach is guided by the step size \(h = 0.05\), offering a detailed means to progress iteratively and closely mimic the behavior of the differential equation step by step.

Differential Equations

Differential equations form the core of modeling change and are equations that involve an unknown function and its derivatives. They are ubiquitous in describing systems ranging from physical phenomenon to biological processes. In this exercise, we deal with an ordinary differential equation given by \(y' = \frac{1+y^2}{1+t}\).

The solution to this equation represents how the function \(y(t)\) changes with respect to \(t\). The derivative \(y'\) indicates how fast or slow \(y(t)\) is changing at any point. Solving such a differential equation provides deeper insights into the dynamics of the modeled process over time. Each additional derivative considered in the Taylor series method adds an extra layer of precision to the approximation, capturing more nuanced aspects of these changes.

Error Analysis

Error analysis is a vital aspect in the application of numerical methods for solving differential equations. It involves understanding the accuracy and precision of an approximation method. In this exercise, part of the task is to evaluate how close the approximated solutions are to the exact solution, which is given as \(y(t) = \tan[\ln(1+t)]\).

This is achieved by calculating the error at \(t=1\), defined as the absolute difference between the exact solution and the Taylor series approximations, for \(p = 1, 2,\) and \(3\). As more terms are added in the series (i.e., increasing \(p\)), the error is expected to decrease, showcasing the convergence of the method towards the exact solution. Utilizing a numerical computation tool can greatly help in performing these calculations efficiently, thus providing a clearer picture of the approximation's reliability.