Problem 9 - Calculus

Question

Consider the solution of the differential equation \(y^{\prime}=y\) passing through \(y(0)=1\) (a) Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,1) (b) Use Euler's method with step size \(\Delta x=0.1\) to estimate the solution at \(x=0.1,0.2, \ldots, 1\) (c) Plot the estimated solution on the slope field; compare the solution and the slope field. (d) Check that \(y=e^{x}\) is the solution of \(y^{\prime}=y\) with \(y(0)=1\)

Step-by-Step Solution

Answer

The exact solution is \( y = e^x \). Euler's method approximates it with increasing accuracy for smaller steps.

Step 1: Identify the Differential Equation

The given differential equation is \( y' = y \). This equation tells us that the rate of change of \( y \) with respect to \( x \) is equal to the value of \( y \).

Step 2: Sketch the Slope Field

To sketch the slope field, note that at each point (x, y), the slope of the solution is y. For example, at y = 1, the slope is 1 (a line with 45-degree slope upward), and at y = 0.5, the slope is 0.5 (less steep). At y = 0, the slope is 0 (horizontal line). Draw these lines at various points.

Step 3: Sketch the Solution Curve

The solution curve passing through the initial condition \( y(0) = 1 \) will be a smooth curve that follows the slope lines we just drew. This is an exponential growth curve starting at (0, 1).

Step 4: Apply Euler's Method Step by Step

Using Euler's method, we start at \( x = 0 \), \( y = 1 \) and take steps of size \( \Delta x = 0.1 \). Compute \( y_{n+1} = y_n + \Delta x \, y'(x_n, y_n) \):- \( y(0.1) = 1 + 0.1 \cdot 1 = 1.1 \)- \( y(0.2) = 1.1 + 0.1 \cdot 1.1 = 1.21 \)- Continue this process up to \( x = 1 \).

Step 5: Calculate Euler's Method Results

Continue calculating using Euler's method:- \( y(0.3) = 1.21 + 0.1 \cdot 1.21 = 1.331 \)- \( y(0.4) = 1.331 + 0.1 \cdot 1.331 = 1.4641 \)- Continue until \( y(1) \).

Step 6: Compare Euler's Method with Exact Solution

The exact solution is \( y = e^x \). Compute \( e^x \) for each step \( x = 0.1, 0.2, \ldots, 1 \) and compare with the estimations from Euler's method. Check the accuracy of the approximation.

Step 7: Verify Analytical Solution

Differentiate \( y = e^x \) to check if it satisfies the DE: \( \frac{d}{dx}(e^x) = e^x \). Also check the initial condition, \( y(0) = e^0 = 1 \), which matches the given condition.

Step 8: Conclude with Comparison

Plot both the Euler's approximation and the exact solution \( y = e^x \) on the slope field. Notice that as step size decreases, Euler's method approximations become closer to the smooth exponential curve of the exact solution.

Key Concepts

Slope Field

A slope field, often called a direction field, is a visual representation of a differential equation without solving it explicitly. For the equation given in the exercise, \( y' = y \), the slope field helps us see how the solution behaves across different points \( (x, y) \) in the plane.

In a slope field, each point has a small line segment indicating the slope, which is obtained from the differential equation. To construct it for \( y' = y \):

• Calculate the slope at several points. For instance, at points where \( y = 1 \), the slope is 1, meaning the line segment tilts upwards at a 45-degree angle.
• If \( y = 0.5 \), the slope is 0.5, showing a less steep tilt.
• At \( y = 0 \), the slope \( y' \) is zero, represented by horizontal line segments.

Drawing these small line fragments creates a field of slopes which visually helps to sketch potential solutions. By matching the lines, you can draw a solution passing through a specific point, such as \( (0, 1) \), generating a broad view of how solutions might behave over a continuum.

Euler's Method

Euler’s Method is a numerical technique used to approximate solutions of differential equations when an exact analytical solution is challenging to obtain. It gives us an iterative approach to estimate the values of \( y \) over small steps in \( x \). It's like walking along the curve in small, straight steps.To apply Euler's Method:

• Start with an initial condition, here \( y(0) = 1 \), and a small step size, such as \( \Delta x = 0.1 \).
• Use the formula \( y_{n+1} = y_n + \Delta x \times y'(x_n, y_n) \) to compute subsequent values.
• Begin at \( x = 0 \) with \( y = 1 \), then calculate: \( y(0.1) = 1 + 0.1 \times 1 = 1.1 \), and \( y(0.2) = 1.1 + 0.1 \times 1.1 = 1.21 \), and so on, up to \( x = 1 \).

With each step, draw a straight segment following the slope direction. The smaller the step size, the closer the approximation to the actual solution, providing a piece-by-piece construction against the exact graph.

Exponential Function

The exponential function, typically written as \( e^x \), arises frequently in natural growth processes and plays a vital role in solving the differential equation \( y' = y \). In this exercise, the anticipated analytical solution follows the equation \( y = e^x \), where each power of \( e \) matches the rate of growth defined by the differential equation.Why Exponential Functions?

• Exponential functions grow continuously, and their derivatives match themselves, making them ideal for equations where a variable's rate of change is proportional to its present value.
• For our equation \( y' = y \), differentiating \( e^x \) results in \( e^x \) again, proving a match with the rate of change \( y' \) expressed.
• Checking the initial condition, with \( y(0) = e^0 = 1 \), the function adequately meets the requirement of passing through \( (0, 1) \).

This exponential function renders not just a closed-form solution, but also visualizes the continuous nature of growth emerging from constant proportional changes.

Analytical Solution

An analytical solution involves a precise and closed-form expression for the solution of a differential equation, as opposed to an approximation like Euler's method. For our simple differential equation \( y' = y \), the goal was to find a function \( y(x) \) that satisfies this differential relationship.Steps to Verify an Analytical Solution:

• Consider the function \( y = e^x \). Differentiate it to find \( y' \), getting \( \frac{d}{dx}e^x = e^x \).
• This satisfies our differential equation, since the derivative \( y' \) equates to \( y \).
• Also confirm the initial condition. Evaluate \( y(0) = e^0 = 1 \), which matches perfectly with \( y(0) \).

By solving analytically, you achieve a direct solution without approximations. In simpler cases, exponential functions like \( y = e^x \) directly reflect the natural expressions of growth and change, making them an integral component of understanding such differential equations.