Question

Euler's Method In Exercises \(73-78\) , use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use \(n\) steps of size \(h .\) $$ y^{\prime}=3 x-2 y, \quad y(0)=3, \quad n=10, \quad h=0.05 $$

Short answer

Euler's method can give an approximate solution of this differential equation. The values calculated using the method for 10 steps would form the solution table.

Step-by-step solution

Understanding the Euler Method Formula

The Euler's method can be formulated as follows: \[y_{i+1} = y_i + h * f(x_i, y_i)\] Here, \(y_{i+1}\) is the next value of the function, \(y_i\) is the current value of the function, \(h\) is the given step size, and \(f(x_i, y_i)\) is the derivative of the function at the current point.

Implementing Euler's method

In this case, the given differential equation is \(y'(x) = 3x - 2y\) and \(y(0) = 3\). Consider this to be our \(f(x, y)\). Therefore we have: \(y_{i+1} = y_i + 0.05 * (3x_i - 2y_i)\). Since \(n=10\), the process should be repeated 10 times with \(x_0 = 0\) and \(y_0 = 3\).

Calculating the approximate solution

On the first iteration \(i=0\), \(x_0 = 0\), \(y_0 = 3\), we substitute into our formula to get the next value \(y_1\): \(y_1 = 3 + 0.05 * (3*0 - 2*3) = 2.7\). The next value \(x_1=x_0 + h = 0 + 0.05 = 0.05\).These calculations should be continued for all remaining 9 steps to get an approximate solution for the differential equation using Euler's method.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In many real-world problems, differential equations represent how quantities change and evolve over time. For example, they can describe population growth, heat transfer, or the motion of objects.

At the core of a differential equation is the idea that the rate of change of a variable is determined by the variable itself. The equation from the exercise, \(y'(x) = 3x - 2y\), is a simple example of a first-order differential equation. It suggests that the rate of change of \(y\), or \(y'\), is dependent on both \(x\) and \(y\) themselves. Solving such equations means finding the function \(y(x)\) that satisfies the equation for all \(x\) in a certain range.

These equations can be challenging to solve analytically, particularly for complex systems. That's why approximation methods like Euler's Method, are important for finding solutions in a practical and efficient way.

Approximate Solutions

Approximate solutions in mathematics are important when exact solutions cannot be easily obtained. Euler's Method is a fundamental technique for finding numerical solutions to differential equations. It provides an approximate solution by iterating over small steps to simulate the behavior of the function.

Here’s how it works:

• Start with an initial value, such as \(y_0=3\) in the given example.
• Use the differential equation to get the next value \(y_1\), applying the method’s formula: \(y_{i+1} = y_i + h * f(x_i, y_i)\).
• Repeat this process for the desired number of steps \(n=10\) in this case, increasing \(x_i\) by a constant step size \(h = 0.05\).

Eventually, this will create a table of \(x, y\) values, representing an approximate curve for the solution in a defined interval. The smaller the step size, the closer the approximation will be to the actual function.

Numerical Methods

Numerical methods are techniques to solve mathematical problems approximately rather than exactly, especially when analytic methods fail or are inconvenient. These methods are essential in fields like engineering, physics, and finance, where they aid in solving complex systems that don't have simple solutions.

Euler's Method is just one example of a numerical method used to solve differential equations. Here's why numerical methods like Euler's Method are valuable:

• They break down complex equations into simpler, solvable iterations.
• They are suitable for computer algorithms, making them ideal for computational modeling.
• They allow for processing and solving real-world problems that involve continuous variables.

Though inherently approximate, numerical methods are powerful because they enable us to predict and understand systems in a quantifiable manner. As we progress in computational abilities, these techniques become indispensable in developing more accurate and comprehensive models.