Question

By the Euler method determine an approximate solution of the initialvalue problem (a) \(y^{\prime}=3 x+y, \quad y(0)=1 ;\) (b) \(y^{\prime}=x^{2}+y, \quad y(0)=2\), at \(x=0.2\) using \(h=0.1\), and compare with the exact solution. Repeat the computations for \(h=0.05\) and \(h=0.2\).

Short answer

Solution: For Part (a) and (b), you will have to calculate the values of \(y_{i+1}\) for each step size and compare them to the exact solution at \(x=0.2\). The step size that gives the closest approximation to the exact solution will be the most accurate. In general, a smaller step size in Euler's method provides a more accurate approximation of the exact solution. However, smaller step sizes require more calculations and can be computationally intensive. The choice of an appropriate step size depends on the desired accuracy and the computation resources available.

Step-by-step solution

Identify the function and initial conditions

For part (a), the given function is \(y^{\prime}=3x+y\), with the initial condition \(y(0)=1\).

Find the exact solution

First, we need to find the exact solution by solving the given differential equation \(y^{\prime}=3x+y\). This is a first-order linear ordinary differential equation, which can be solved using an integrating factor. The general solution can be written as: \(y(x) = e^{-x}(C + \int e^{x}(3x)dx)\). Solving the integral and using the initial condition \(y(0)=1\), we find the exact solution to be \(y(x)=e^{-x}(2-x)\).

Apply Euler's method with different step sizes

We will now apply Euler's method with three different step sizes: \(h=0.1\), \(h=0.05\), and \(h=0.2\). For each step size, we will use the following equation to find the approximate value \(y_{i+1}\): \(y_{i+1} = y_i + h(3x_i + y_i)\) Compute the values of \(y_{i+1}\) using the given step sizes and compare with the exact solution at \(x=0.2\). #Part (b)#

Identify the function and initial conditions

For part (b), the given function is \(y^{\prime}=x^2+y\), with the initial condition \(y(0)=2\).

Find the exact solution

First, let's find the exact solution by solving the given differential equation \(y^{\prime}=x^2+y\). This is also a first-order linear ordinary differential equation, which can be solved using an integrating factor. The general solution can be written as: \(y(x) = e^{-x}(C + \int e^{x}(x^2)dx)\). Solving the integral and using the initial condition \(y(0)=2\), we find the exact solution to be \(y(x) = e^{-x}(1 + x + x^2)\).

Apply Euler's method with different step sizes

Again, we will apply Euler's method with three different step sizes: \(h=0.1\), \(h=0.05\), and \(h=0.2\). For each step size, we will use the following equation to find the approximate value \(y_{i+1}\): \(y_{i+1} = y_i + h(x^2_i + y_i)\) Compute the values of \(y_{i+1}\) using the given step sizes and compare with the exact solution at \(x=0.2\).

Key concepts

Initial Value Problem

An initial value problem is a type of differential equation coupled with a specified value at a point, known as the initial condition. Consider the expression:

• The form: \( y' = f(x, y) \) is a differential equation describing how \( y \) changes with \( x \).
• The initial condition might be something like \( y(x_0) = y_0 \), which gives the starting value.

This allows us to solve the differential equation in a more specific way to find a unique solution that satisfies the initial condition.
In our example, the initial value problems include differential equations such as \( y' = 3x + y \) with \( y(0) = 1 \), and \( y' = x^2 + y \) with \( y(0) = 2 \). Both of these conditions act as a starting point when we use numerical methods like Euler’s method to approximate solutions.

First-Order Linear Ordinary Differential Equations

First-order linear ordinary differential equations are among the simplest types of differential equations. They are written in the form:

• \( y' = P(x) y + Q(x) \)

These equations involve derivatives to the first degree and no higher, which means they describe phenomena with a single rate of change. Solving them typically involves using the integrating factor method.
For instance, in our exercises, both differential equations \( y' = 3x + y \) and \( y' = x^2 + y \) conform to this format. The solving process entails finding an integrating factor to simplify the differential equation, making integration straightforward. Using an integrating factor, like \( e^{-x} \), allows one to find the exact analytical solutions, which can then be evaluated to understand results from numerical methods like Euler’s method.

Step Size in Numerical Methods

Step size plays a crucial role in numerical methods, particularly when using the Euler’s method to approximate solutions of differential equations. Consider the following points:

• Step Size \( h \): It dictates how large each step is from one approximation to the next.
• Effect on Accuracy: Smaller step sizes \( h \) yield more accurate approximations by closely following the curve of the exact solution, but require more computations.
• For example: In the exercise, step sizes of \( h = 0.1, 0.05, \) and \( 0.2 \) were used to demonstrate how each impacts the computed solution.

In practical terms, a balance between accuracy and computational effort must be found. Using smaller steps increases precision and convergence towards the exact solution, offering more reliable results. Comparing the results for different values of step size at a specific point, such as \( x = 0.2 \), offers a direct view of how step size affects the approximation quality.