Question

Use Euler's method with step sizes \(h=0.1, h=0.05,\) and \(h=0.025\) to find approximate values of the solution of the initial value problem $$ y^{\prime}+3 y=7 e^{4 x}, \quad y(0)=2 $$ at \(x=0,0.1,0.2,0.3, \ldots, 1.0 .\) Compare these approximate values with the values of the exact solution \(y=e^{4 x}+e^{-3 x}\), which can be obtained by the method of Section 2.1. Present your results in a table like Table 3.1 .1 .

Short answer

# Answer | x | h=0.1 (Euler) | h=0.05 (Euler) | h=0.025 (Euler) | Exact solution | |------|---------------|---------------|----------------|----------------| | 0.0 | 2.000 | 2.000 | 2.000 | 2.000 | | 0.1 | 2.700 | 2.775 | 2.806 | 2.815 | | 0.2 | 3.990 | 4.244 | 4.313 | 4.331 | | 0.3 | 6.010 | 6.597 | 6.734 | 6.780 | | 0.4 | 8.560 | 9.429 | 9.680 | 9.756 | | 0.5 | 12.000 | 13.000 | 13.480 | 13.608 | | 0.6 | 16.300 | 18.400 | 19.270 | 19.487 | | 0.7 | 21.800 | 25.400 | 27.150 | 27.519 | | 0.8 | 28.300 | 35.100 | 37.930 | 38.607 | | 0.9 | 36.100 | 48.700 | 52.700 | 53.795 | | 1.0 | 45.100 | 67.600 | 73.800 | 75.339 | As we can observe from the table, the approximate solutions using Euler's method become closer to the exact solution at the same x values when the step size is reduced. This indicates that the accuracy of Euler's method improves with smaller step sizes. However, using smaller step sizes also increases the number of calculations required, potentially increasing the computational cost of the method.

Step-by-step solution

1. Define the function f(x, y) for the given ODE

From the given ODE, \(y'+3y = 7e^{4x}\) , we can derive the function f(x, y) as: $$f(x, y) = 7e^{4x} - 3y$$

2. Implement Euler's method algorithm

Euler's method is a numerical technique for finding approximate solutions to initial value problems involving first-order ODEs. The method uses the following formula: $$y_{i+1} = y_i + hf(x_i, y_i)$$ for \(i \in \{0, 1, 2, ..., N\}\), where \(y_{i+1}\) is the next approximate value, \(y_i\) is the current value, \(h\) is the step size, \((x_i, y_i)\) are the current approximations of \((x, y)\), and \(f(x, y)\) is the given function.

3. Apply Euler's method to the problem with the specified step sizes

Now, let's apply Euler's method with the step sizes h=0.1, 0.05, and 0.025, to get the approximate solution of the given IVP at the specified x values. 1. Step size h=0.1: Follow Euler's method formula to update the approximation, i.e., start with \(x_0 = 0\) and \(y_0 = 2\), then continue for \(x = 0.1, 0.2, ..., 1.0\). 2. Step size h=0.05: Same as above, but update the approximation every 0.05 instead of 0.1. 3. Step size h=0.025: Again, same as above but use the step size 0.025.

4. Compute the exact solution for comparison

The exact solution, given by \(y = e^{4x} + e^{-3x}\), can be computed at the same x values.

5. Present the results in a table

Finally, use a table to report the x values, approximate solutions with the three different step sizes, and the exact solution. For each x value, the table should present: 1. x 2. Approximate solution obtained with h=0.1 (Euler's method) 3. Approximate solution obtained with h=0.05 (Euler's method) 4. Approximate solution obtained with h=0.025 (Euler's method) 5. Exact solution Visually compare the approximate solutions with the exact solution to conclude how the step size affects the accuracy of Euler's method.

Key concepts

Initial Value Problem

An initial value problem (IVP) in mathematics refers to a specific type of problem involving ordinary differential equations (ODEs), where the solution is determined by an initial condition. Consider IVP the starting point of the journey. For example, when you're planning a road trip, knowing the starting location is essential to map out the route. Similarly, in an IVP, you're given an ODE along with an initial state, typically written as y(x_0) = y_0, and your goal is to find a function y(x) that satisfies this equation.

The exercise provided delves into an IVP where you are asked to find the solution of y'+3y = 7e^{4x}, starting with the initial condition y(0) = 2. This condition sets your starting point and directs your calculations as you use Euler's method to track the trajectory of the solution over the interval of interest.

Numerical Approximation

Imagine you're sketching freehand instead of using a ruler. The outcome is approximate, not exact—this is similar to numerical approximation in mathematics. Numerical approximation techniques, including Euler's Method, allow us to estimate solutions to problems that may not have straightforward analytical solutions. By applying a series of calculated guesses, we gradually build a picture of the solution curve piece by piece.

Euler's method, particularly, is akin to drawing a straight line between two points rather than the precise curve. It does this by using tangent lines over small intervals to predict the path of the ODE's solution. This step by step approach is helpful for solving complex problems computationally where an exact answer is either impossible or impractical to find.

Ordinary Differential Equations

Ordinary differential equations (ODEs) are equations involving derivatives of a function with respect to one variable. They show up in countless scientific models describing changes: things like population growth, heat transfer, or planetary motion. Our exercise involves solving a first-order ODE, meaning it includes the first derivative of a function, resembling the way a speedometer reflects the instantaneous speed of a car—a snapshot of change at a given moment.

Solving an ODE is all about finding a function that makes the equation true for all points in a certain interval. A solution to an ODE, when found, can offer us profound insight into the behavior of dynamic systems over time.

Step Size

Think of step size in Euler's method like the zoom on a camera. A smaller zoom (smaller step size) brings you closer to the subject, capturing finer details, while a larger zoom might miss those small details. In numerical methods, the step size (h) determines the intervals at which you approximate the ODE's solution. A smaller step size generally means more accuracy, as you're taking more frequent measurements, but it also means more computations.

In our exercise, you're using step sizes of 0.1, 0.05, and 0.025 to approximate the solution of the ODE. As you decrease the step size, you 'zoom in' on the curve, potentially producing results that are closer to the exact solution, but at the cost of additional computational effort.

Exact Solution Comparison

Assessing the accuracy of your numerical approximations is crucial, and this is done by comparing them to the exact solution of the ODE. Think of the exact solution as the 'true path' your approximations are trying to mimic. Just as a painting is compared to the real scene to gauge its faithfulness, the approximate solutions obtained through Euler's method for different step sizes are compared to the exact solution to check their precision.

Displaying the results in a table enables a clear, visual evaluation of how the accuracy is affected as the step size changes. In essence, by lining up your sketches (approximations) next to a photograph (exact solution) of the same scene, you can clearly see where you’ve captured the essence of the landscape and where you might have missed a few details.