Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=x, y(0)=0,[0,1] $$

Step-by-Step Solution

Answer

Using Euler's Method, the approximation of y at x = 1 is 0.60.

Step 1: Understanding Euler's Method

Euler's Method is a numerical technique to approximate solutions of differential equations. Given the differential equation \(y' = f(x, y)\), it estimates \(y\) by updating it iteratively: \(y_{n+1} = y_n + hf(x_n, y_n)\), where \(h\) is the step size.

Step 2: Setting Initial Conditions

We start with the initial condition given in the problem, \(y(0) = 0\), and the initial value of \(x\) as 0. The step size \(h = 0.2\) and we need to approximate \(y\) over the interval \([0,1]\).

Step 3: First Iteration

Using Euler's formula, calculate \(y(0.2): y_1 = y_0 + h \, f(x_0, y_0) = 0 + 0.2 \times 0 = 0\). Now, update \(x_1 = x_0 + h = 0.2\).

Step 4: Second Iteration

For \(x = 0.2\), compute \(y_2: y_1 + h \, f(x_1, y_1) = 0 + 0.2 \times 0.2 = 0.04\). Update \(x_2 = 0.2 + 0.2 = 0.4\).

Step 5: Third Iteration

For \(x = 0.4\), calculate \(y_3: y_2 + h \, f(x_2, y_2) = 0.04 + 0.2 \times 0.4 = 0.12\). Update \(x_3 = 0.4 + 0.2 = 0.6\).

Step 6: Fourth Iteration

For \(x = 0.6\), obtain \(y_4: y_3 + h \, f(x_3, y_3) = 0.12 + 0.2 \times 0.6 = 0.24\). Update \(x_4 = 0.6 + 0.2 = 0.8\).

Step 7: Fifth Iteration

For \(x = 0.8\), calculate \(y_5: y_4 + h \, f(x_4, y_4) = 0.24 + 0.2 \times 0.8 = 0.40\). Update \(x_5 = 0.8 + 0.2 = 1.0\).

Step 8: Sixth Iteration

Finally, for \(x = 1.0\), compute \(y_6: y_5 + h \, f(x_5, y_5) = 0.40 + 0.2 \times 1.0 = 0.60\). This is the approximation at \(x = 1.0\).

Key Concepts

Numerical Methods

Numerical methods are like mathematical recipes that help us tackle problems when an exact answer is hard or impossible to find analytically. These methods are especially useful in scenarios involving complex equations or situations where exact solutions are difficult. By focusing on approximate solutions, numerical methods provide us with practical means to solve problems, like those found in real-world applications.

• **Efficiency**: They allow computations when formulas are too complex.
• **Accuracy**: With each method, a balance between accuracy and speed is maintained.
• **Versatility**: They can help solve various types of equations, such as algebraic or differential equations.

Euler's Method is one of these widely used numerical techniques because of its simplicity and ease of implementation.

Differential Equations

Differential equations are mathematical equations that involve derivatives. A derivative tells us how a function changes at any given point, providing crucial insight into dynamic systems.

• **Ordinary Differential Equation (ODE)**: This is where there is a relationship between functions of one variable and its derivatives, like the equation we see in the exercise: \( y' = x \).
• **Applications**: These equations model scenarios such as population growth, heat conduction, and financial markets.
• **Challenges**: Solving them analytically can be difficult or impossible, making numerical methods indispensable tools.

Euler's Method is specifically employed to approximate solutions for ordinary differential equations.

Initial Value Problems

An initial value problem involves finding a solution to a differential equation subject to specified values at a starting point. These problems are essential in understanding how a system evolves over time, starting from an initial state.

• **Initial Conditions**: For the problem in question, the initial condition is \( y(0) = 0 \).
• **Purpose**: They help us predict the future behavior of systems described by differential equations.
• **Relevance**: Initial conditions are critical because they determine the specific solution out of potentially many possibilities.

Euler's Method begins with given initial conditions and iteratively approximates the solution over the desired interval.

Step Size

When using numerical methods like Euler's Method, step size, denoted by \( h \), is a crucial factor. It determines the intervals at which calculations are made, thus affecting both the accuracy and efficiency of the solution.

• **Significance**: A smaller step size usually offers more accurate results but requires more computations. For instance, in the exercise, the chosen step size is \( h = 0.2 \).
• **Trade-off**: Larger step sizes make calculations faster but can lead to errors and inaccuracies.
• **Implementation**: Controlling step size is one of the easiest ways to adjust the precision of an iterative method.

Thus, choosing the right step size is a balance of computational resources and desired precision.

Approximation Techniques

Approximation techniques are strategies to find solutions that are close enough to the exact ones, especially when the exact solutions are either difficult or impossible to find. Euler's Method serves as a straightforward approximation method for solving differential equations.

• **Purpose**: Provide approximate solutions that inform decision-making when exact answers are impractical.
• **Process**: Euler’s Method simplifies this by calculating step-by-step, with each step aiming to minimize the error.
• **Applications**: Widely used in scientific computations to model behaviors that require solutions to differential equations.

Such methods are invaluable in fields where mathematical models predict real-world behaviors, and accuracy is gradually refined through iterations.