Question

Use Euler's Method with \(n=5\) steps over the interval \(t=[0,1] .\) Then solve the initial-value problem exactly. How close is your Euler's Method estimate? $$ y^{\prime}=3^{x}-2 y, y(0)=0 $$

Short answer

Euler's Method estimate is 1.1928, exact solution is approximately 1.0677, differing by 0.1251.

Step-by-step solution

Define the Euler's Method and Interval

Euler's Method is a numerical technique for solving ordinary differential equations (ODEs) with a given initial value. It approximates the solution by using a series of steps. Here, we have an interval from \(t=0\) to \(t=1\) with \(n=5\) steps. The step size \(h\) is calculated as \(h = \frac{1-0}{5} = 0.2\).

Compute Euler's Method Approximations

We start with the initial condition \(y(0) = 0\) and use the relation \(y_{n+1} = y_n + h(y'_{n})\), where \(y' = 3^x - 2y\). Calculate the approximate values for each step:- \(y_1 = y_0 + 0.2(3^0 - 2\cdot0) = 0 + 0.2 \cdot 1 = 0.2\)- \(y_2 = y_1 + 0.2(3^{0.2} - 2\cdot0.2) \approx 0.2 + 0.2 \cdot 0.972 = 0.3944\)- \(y_3 = y_2 + 0.2(3^{0.4} - 2\cdot0.3944) \approx 0.3944 + 0.2 \cdot 1.192 = 0.6328\)- \(y_4 = y_3 + 0.2(3^{0.6} - 2\cdot0.6328) \approx 0.6328 + 0.2 \cdot 1.351 = 0.9030\)- \(y_5 = y_4 + 0.2(3^{0.8} - 2\cdot0.9030) \approx 0.9030 + 0.2 \cdot 1.449 = 1.1928\)Thus, the Euler's approximation at \(t = 1\) is \(y_5 \approx 1.1928\).

Solve the Initial-Value Problem Exactly

To find the exact solution, we solve the differential equation \(y' = 3^x - 2y\). This is a first-order linear differential equation that can be solved using the integrating factor method. The integrating factor is \(e^{-2x}\), and the solution is constructed as follows:\[y(t) = e^{2x} \int e^{-2x}3^x \, dx + Ce^{-2x} \]The integration \(\int e^{-2x}3^x \, dx\) can be solved using integration by parts or by recognizing patterns, leading to:\[y(x) = \frac{3^x}{\ln(3) + 2} + Ce^{-2x}\]Substitute the initial condition \(y(0) = 0\), which gives \(C = -\frac{1}{\ln(3) + 2}\). Therefore, the exact solution is:\[y(x) = \frac{3^x}{\ln(3) + 2} - \frac{e^{-2x}}{\ln(3) + 2}\]At \(t = 1\), calculate:\[y(1) = \frac{3}{\ln(3) + 2} - \frac{e^{-2}}{\ln(3) + 2} \approx 1.0677\]

Compare Results

Compare Euler's method approximation with the exact solution. Euler's approximation at \(t = 1\) is \(1.1928\), while the exact solution is approximately \(1.0677\). The difference is \(1.1928 - 1.0677 = 0.1251\). Euler's Method overestimates the exact solution by approximately \(0.1251\).

Key concepts

Numerical Methods

Numerical methods are techniques used to approximate solutions for mathematical problems that do not have simple analytical solutions. These methods are essential in solving various problems in engineering, physics, and other sciences where exact solutions are difficult or impossible to find. A common numerical method is Euler's Method, which is used for approximating solutions to ordinary differential equations (ODEs). It relies on step-by-step iterations to estimate the value of a function at a given point based on initial conditions.

Advantages of numerical methods include their ability to handle complex problems with non-linear equations. They provide a way to obtain solutions even if the differential equations cannot be solved with traditional integration techniques. Despite being approximations, these methods can give insight into the behavior of the solution, especially over small intervals where the error can be controlled.

Ordinary Differential Equations

Ordinary differential equations (ODEs) are equations that involve functions of one independent variable and their derivatives. These equations are vital since they describe various phenomena in natural sciences, engineering, and economics.

In the exercise example, the differential equation is given by: \[ y^{\prime} = 3^{x} - 2y \]This is a first-order linear ODE, indicating that the highest derivative is one and involves linear combinations of the function and its derivatives. Solving such equations involves finding a function that satisfies the equation given a set of initial conditions.

Understanding ODEs is crucial because they model dynamic systems and processes, such as motion, growth of populations, and electrical circuits. Each type of ODE may require different methods for finding solutions, such as separation of variables, integrating factors, or numerical techniques like Euler's Method.

Initial Value Problem

An initial value problem (IVP) consists of an ordinary differential equation along with specified values at the start of the interval of interest. These initial conditions allow us to find a unique solution to the differential equation.

In the given exercise, the initial value problem involves:

• The equation: \[ y^{\prime} = 3^{x} - 2y \]
• The condition: \[ y(0) = 0 \]

Solving an IVP means determining a function that not only satisfies the differential equation but also passes through the initial starting point. This ensures specificity in the solution, making it applicable to real-world problems where initial states are known.
Initial value problems appear frequently in applications requiring prediction or modeling from known starting conditions, such as reaction kinetics in chemistry or investment growth in finance.

Integration Techniques

Integration techniques are methods used to find the antiderivatives or integrals of functions. These are essential tools in solving differential equations, including those encountered in initial value problems.

In the context of the given exercise, integration techniques come into play when solving the exact solution of the ODE: \[ y(x) = \frac{3^x}{\ln(3) + 2} - \frac{e^{-2x}}{\ln(3) + 2} \]This involves finding an integrating factor to facilitate the integration process. Techniques like integration by parts or recognizing standard integral forms are useful here.

Mastery of integration techniques allows students to solve a wide array of differential equations analytically. These techniques can reveal deeper insights into the problem, such as long-term behavior and stability of solutions, providing a more complete understanding beyond numerical approximations.