Problem 16 - Calculus Early Transcendentals

Question

(a) Show that the solution of the initial-value problem \(y^{\prime}=e^{-x^{2}}, y(0)=0\) is $$ y(x)=\int_{0}^{x} e^{-t^{2}} d t $$ (b) Use Euler's Method with \(\Delta x=0.05\) to approximate the value of \(y(1)=\int_{0}^{1} e^{-t^{2}} d t\) and compare the answer to that produced by a calculating utility with a numerical integration capability.

Step-by-Step Solution

Answer

Euler's Method approximates numerical results, but direct numerical integration from a calculating utility gives a more accurate value of \( y(1) = \int_0^1 e^{-t^2} dt \).

Step 1: Understanding the Differential Equation

We begin by recognizing that the given differential equation is \( y' = e^{-x^2} \) with an initial condition \( y(0) = 0 \). This is a first-order ordinary differential equation which can be solved by integration.

Step 2: Solving the Differential Equation

To solve \( y' = e^{-x^2} \), we integrate the right-hand side with respect to \( x \) to find \( y(x) \). This gives us: \[ y(x) = \int e^{-x^2} \, dx = \int_0^x e^{-t^2} \, dt \] This expression satisfies the initial condition \( y(0) = 0 \) since \( \int_0^0 e^{-t^2} \ dt = 0 \). Therefore, \( y(x) = \int_0^x e^{-t^2} dt \) is the solution.

Step 3: Setting Up Euler’s Method

Euler's Method approximates solutions to differential equations using stepwise predictions. Here, the step size is \( \Delta x = 0.05 \). From the initial condition, we know \( y(0) = 0 \), and we need to approximate \( y(1) \). We will use the formula: \[ y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n) \] where \( f(x, y) = e^{-x^2} \).

Step 4: Iterating Euler's Method

Start with \( y_0 = 0 \), \( x_0 = 0 \). For each step, compute: * \( y_1 = y_0 + 0.05 \cdot e^{-0^2} = 0 + 0.05 \cdot 1 = 0.05 \)* Optimize steps for \( x = 0.05 \) to \( x = 1.00 \) by incrementing \( x \) by 0.05, updating \( y \) at each step using the formula above.

Step 5: Calculating with Numerical Integration

To compare, use a numerical integration tool (e.g., Simpson's Rule, Trapezoidal Rule, or a calculator) to compute \( \int_0^1 e^{-t^2} \, dt \) and verify its result against the approximation obtained from Euler's Method steps.

Step 6: Comparing the Results

Compare the result obtained through Euler’s Method to the accurate numerical integration from a calculator. Euler's Method will give an approximate value, generally less accurate due to step size differences or rounding.

Key Concepts

Euler's Method

Euler's Method is a straightforward numerical technique for solving ordinary differential equations (ODEs) with a given initial value, known as an initial value problem. We use it to approximate the solution by breaking the range into small steps. The main idea is to use the slope at the beginning of each interval to predict the value of the function at the end of that interval.
When using Euler's Method, we increase the variable in small increments (denoted as \( \Delta x \) or \( h \)), updating the current estimate of the solution using the formula:

• \( y_{n+1} = y_n + h \cdot f(x_n, y_n) \)

Here, \( y_n \) is the current solution, \( x_n \) is the current x-value, and \( f(x_n, y_n) \) is the function's derivative.
In simpler terms, you evaluate the current slope, estimate how much your function should change over a tiny step, and update your value accordingly. Euler's Method is simple, but its accuracy depends on the step size \( h \). Smaller steps yield more accurate predictions but require more computations.

Numerical Integration

Numerical integration is a process to approximate the value of an integral, especially when an analytic solution is difficult or impossible to find. It's particularly useful in solving definite integrals over a specific interval.The process involves several methods, the most common being:

• Simpson's Rule: Uses parabolic approximations over small subintervals to improve accuracy.
• Trapezoidal Rule: Approximates the area under the curve by breaking it into trapezoids.
• Rectangular Methods: Simplest form where the area is split into rectangles.

Each of these methods provides a systematic approach for estimating integrals to suitable precision compared to directly solving integral equations.
In the exercise, one such numerical method could be used to compute \( \int_{0}^{1} e^{-t^2} \, dt \), offering a reference to check the approximation from Euler's Method. Such comparison shows the strength of numerical integration as it often yields more precise results because it considers the function's entire behavior within the interval.

Initial Value Problem

An Initial Value Problem (IVP) is a type of differential equation where the solution is required to meet specific conditions at the start (initial conditions). It involves solving a differential equation, \( f'(x) = g(x, y) \), with a given value \( y(x_0) = y_0 \).The initial condition helps uniquely determine which among the potentially infinite solutions to the differential equation is the correct one, fitting the real-world context you are modeling.
In the context of our exercise, the problem starts with the differential equation \( y' = e^{-x^2} \) and initial condition \( y(0) = 0 \). This frames it as an IVP because it combines the general solution of the ODE with the specific condition \( y(0) = 0 \). Solving it involved integrating the given formula to find a function \( y(x) \) that starts passing through \( (0, 0) \), leading to \( y(x) = \int_0^x e^{-t^2} dt \), which satisfies the initial condition while providing a functional solution for \( y \).