Problem 53 - Calculus: Graphical, Numerical, Algebraic

Question

Percentage Error Let \(y=f(x)\) be the solution to the initial value problem \(d y / d x=2 x+1\) such that \(f(1)=3 .\) Find the per- centage error if Euler's Method with \(\Delta x=0.1\) is used to approxi- mate \(f(1.4) .\)

Step-by-Step Solution

Answer

The percentage error if Euler's Method with \( \Delta x = 0.1 \) is used to approximate \( f(1.4) \) from the initial value problem \( \frac{dy}{dx}=2x+1 \) with \( f(1)=3 \) is approximately 5.29%.

Step 1: Calculate the true value

Firstly, an analytical solution to the differential equation needs to be calculated. The differential equation \( \frac{dy}{dx}=2x+1 \) is linear and straightforward to solve: integrating gives \( y=x^2+x+c \). The constant \( c \) is found by substituting \( x =1 \) and \( y = 3 \) (from the condition \( f(1) = 3 \)): \( c = 3-1^2 -1 = 1 \). Thus, the analytical solution is \( y=x^2+x+1 \). Finally, substituting \( x=1.4 \) gives the true value as \( y=1.4^2 + 1.4 + 1 = 4.16 \).

Step 2: Apply Euler's Method

Applying Euler's method for \( \Delta x = 0.1 \), the initial conditions are \( x_0 =1 \), \( y_0=3 \). Then iteratively compute: \( y_{n+1} = y_n + \Delta x * f(x_n, y_n) \), where \( f(x, y) = 2x + 1 \). The x-values will be 1, 1.1, 1.2, 1.3, and finally 1.4, and for each compute the corresponding y-value. After performing the calculations, you will find that the approximate value at x=1.4 will be 3.94.

Step 3: Calculate the Percentage Error

Finally, compute the percentage error with the formula \( \frac{|true value - approx value|}{|true value|} * 100\%. \) Substituting the true value as 4.16 and the approximate value from Euler's method as 3.94, the result is approximately 5.29%.

Key Concepts

Percentage Error in Numerical Methods

Understanding the percentage error in numerical methods is essential to gauge the accuracy of approximations, especially when solving problems that require iterative methods such as Euler's Method. The percentage error is a measure that compares the approximate value obtained through a numerical method to the exact (true) value. It's calculated using the formula:
\[ \text{Percentage Error} = \frac{|\text{true value} - \text{approx value}|}{|\text{true value}|} \times 100\% \]
When applying Euler's Method for an initial value problem, we often deal with slight errors in each step that compound over successive iterations. It's crucial to keep the step size, \( \Delta x \), small to minimize these errors. However, trade-offs between computational time and accuracy must be considered.

Improving Accuracy

To reduce percentage error in numerical solutions, one might:

• Decrease the step size (\( \Delta x \)) if computationally feasible.
• Increase the complexity of the numerical method used (e.g., using higher-order methods).
• Employ error correction techniques after initial approximations.

These measures aim to provide a more precise approximation closer to the true value.

Initial Value Problem

An initial value problem is a specific type of differential equation, combined with a set of conditions, that defines a unique solution. It typically involves finding a function \( y \) that satisfies a differential equation \( \frac{dy}{dx} = f(x, y) \), and that passes through a given point, called the initial condition \( (x_0, y_0) \). The goal is to determine this function over some interval.In the context of our exercise, we are given the initial value problem: \[ \frac{dy}{dx} = 2x + 1 \
y(1) = 3 \
\]We then use this initial value to generate the subsequent points using Euler's Method, which provides an algorithmic approach to approximate the solution curve of the differential equation. Initial value problems are foundational in various scientific fields, from physics to finance, as they often model time-dependent processes where initial conditions are known.

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They're powerful tools for modeling the way systems change over time and are used extensively across disciplines such as physics, engineering, and economics. The key to solving a differential equation is to find the unknown function that satisfies the relationship expressed by the equation.

Linear vs. Nonlinear

Differential equations can be classified into linear and nonlinear. Linear differential equations, like the one in our exercise \( \frac{dy}{dx}=2x+1 \), have solutions that can be easily found through integration and involve a sum of terms, each with a constant or function multiplied by the unknown function or its derivatives.Nonlinear equations, on the contrary, involve terms with powers or products of the unknown function and its derivatives. They are typically more challenging to solve, and often require specialized techniques or numerical methods for their solutions.Solving a differential equation can provide insights into the behavior of complex systems by representing the change in terms of the system's current state. The result, whether an exact solution or a numerical approximation, yields a function that models the behavior of the system over time.