Question

Use Euler's method and the Euler semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval. $$ y^{\prime}+3 y=x y^{2}(y+1), \quad y(0)=1 ; \quad h=0.1,0.05,0.025 \text { on }[0,1] $$

Short answer

#Question# Apply Euler's method and Euler Semilinear method for the given initial value problem with different step sizes, and compare their solutions at 11 equally spaced points in the interval [0,1]. The initial value problem is: $$ y^{\prime}+3 y=x y^{2}(y+1), \quad y(0)=1 $$

Step-by-step solution

Understand the formulas for Euler's Method and Euler Semilinear Method

For Euler's method, given a differential equation of the form \(y'(x) = f(x,y)\) and an initial condition \(y(x_0) = y_0\), we can approximate the solution using the formula: $$ y_{n+1} = y_n + hf(x_n, y_n) $$ For Euler's semilinear method, given a semilinear differential equation of the form \(y'(x) + ky(x)=f(x,y)\), we can approximate the solution using the formula: $$ y_{n+1} = y_n \frac{1+(h/2)k+hf(x_n,y_n)}{1-(h/2)k} $$

Apply Euler's method for different step sizes

We have \(f(x,y)=-3y+xy^2(y+1)\), and the given initial condition is \(y(0)=1\). We will now apply Euler's method for the given step sizes to find the approximate values at 11 equally spaced points in the interval [0,1]. - For \(h = 0.1\), we have 10 intervals, and the points \(x_i = 0, 0.1, 0.2, ..., 1\). - For \(h = 0.05\), we have 20 intervals, and the points \(x_i = 0,0.05, 0.1, ..., 1\). - For \(h = 0.025\), we have 40 intervals, and the points \(x_i = 0,0.025,0.05, ..., 1\). Apply Euler's method for the different step sizes and find approximate values at the required points.

Apply Euler Semilinear method for different step sizes

We have the semilinear differential equation of the form \(y'(x) + 3y(x)= x y^2(y+1)\). Apply the Euler Semilinear method for the given step sizes to find approximate values at 11 equally spaced points in the interval [0,1]. - For \(h = 0.1\), we have 10 intervals, and the points \(x_i = 0, 0.1, 0.2, ..., 1\). - For \(h = 0.05\), we have 20 intervals, and the points \(x_i = 0,0.05, 0.1, ..., 1\). - For \(h = 0.025\), we have 40 intervals, and the points \(x_i = 0,0.025,0.05, ..., 1\). Apply Euler Semilinear method for the different step sizes and find approximate values at the required points.

Compare the solutions of Euler's Method and Euler Semilinear Method

After finding the approximate values for both methods, compare the results at each step size and observe the difference in their performance. It is expected that Euler Semilinear Method will provide better accuracy than Euler's Method for the given initial value problem, especially with increasing step sizes.

Key concepts

Euler Semilinear Method

In understanding the dynamics of a differential equation, particularly semilinear ones, the Euler Semilinear Method is a noteworthy tool. This method is a revision of the general Euler's Method, tailored specifically for equations where the function can be separated into linear and nonlinear parts. Applied to our differential equation \(y' + 3y = xy^2(y + 1)\), the method enables us to approximate the solution iteratively. The semilinear approach modifies the updating rule, so it becomes particularly effective when dealing with such problems. To use this method, one applies the formula:
\[ y_{n+1} = y_n \frac{1+(h/2)k+hf(x_n,y_n)}{1-(h/2)k} \]
where \( h \) is the step size, and \( k \) is the coefficient of the linear term. By dividing the interval into smaller segments, the method updates the value of \( y \) at each step, factoring in both the linear and the nonlinear contribution, hence often yielding more accurate results compared to traditional Euler's Method.

Initial Value Problem

At the heart of many differential equation problems lies the Initial Value Problem (IVP). This is a situation where you're given a differential equation along with an initial condition—the starting value of the unknown function at a specific point. For our exercise, the IVP is defined by the differential equation \(y' + 3y = xy^2(y + 1)\) with the initial condition \(y(0) = 1\). Solving an IVP means finding the unique function that not only satisfies the differential equation but also passes through the initial point provided. It's like plotting a path knowing exactly where you start. Both Euler's Method and the Euler Semilinear Method start with this initial value to generate a sequence of approximations that, ideally, converge towards the actual solution of the differential equation.

Numerical Approximation

When analytic solutions to differential equations are difficult or impossible to find, numerical approximation methods like Euler's Method come into play. These techniques allow us to estimate the solution's behavior by constructing a series of values that approximate the true solution at discrete points. They work by 'stepping' through the problem domain, jumping from one point to the next and using information from the previous step to predict the next value. This process transforms the continuous problem of solving differential equations into a manageable discrete one. In our exercise, numerical approximation provides a practical way to understand the behavior of the solution y(x) across the interval [0,1]. It's akin to sketching a complex shape by plotting individual dots and connecting them to see the overall figure.

Step Size

The concept of 'step size', denoted as \( h \), is crucial in numerical methods for differential equations. In our exercise, step size directly impacts the granularity of our approximation—the smaller the step, the finer the detail we can capture. However, with finer steps come more calculations, necessitating a balance between computational effort and the accuracy desired. The step size choices of \( h = 0.1, 0.05, 0.025 \) offer varying levels of precision, with \( h = 0.025 \) potentially giving the closest approximation to the true solution. It's important to select an appropriate step size; too large, and the approximation may miss important features of the solution, too small, and the computation may become impractical.

Differential Equation Solution

Ultimately, the goal of employing methods such as Euler's and Euler Semilinear is to solve a differential equation or, more realistically, to approximate its solution. In our exercise, we're investigating the equation \(y' + 3y = xy^2(y + 1)\), for which an exact solution may not be easily obtainable. By using the given methods, we step closer to understanding how \( y \) behaves as \( x \) changes from 0 to 1. While the methods yield numerical approximations rather than precise formulas, these approximations provide invaluable insights into the tendencies of the solution, thus allowing students and researchers to make informed predictions about the system that the differential equation models. With sufficient refinement, such as reducing step size, these approximations can become quite precise, offering a window into the equation's true character.