Question

Use the improved Euler method to find approximate values of the solution of the given initial value problem at the points \(x_{i}=x_{0}+i h,\) where \(x_{0}\) is the point where the initial condition is imposed and \(i=1,2,3\). $$ y^{\prime}=\frac{1+x}{1-y^{2}}, \quad y(2)=3 ; \quad h=0.1 $$

Short answer

Question: Using the improved Euler method, find the approximate values of the solution of the given initial value problem \(y'\) = \(\frac{1+x}{1-y^2}\), y(2) = 3, h = 0.1, at the points \(x_i = x_0 + ih\), for \(i = 1, 2, 3\). Answer: After applying the improved Euler method, the approximate values of the solution for \(i = 1, 2, 3\) are \(y_1 = \frac{9801}{3260}\), \(y_2 \approx 6.183\), and \(y_3 \approx 6.465\).

Step-by-step solution

Set up initial values

We need to determine the initial values based on the given initial value problem. \(x_0 = 2\), \(y_0 = 3\), and the step size \(h = 0.1\).

Calculate the Euler prediction

With initial values in hand, we can calculate the Euler prediction. The Euler prediction can be calculated using the formula: \(y_{1,predicted} = y_0 + h f(x_0, y_0)\), where f(x, y) is the function given in the initial value problem. First, substitute the initial values into the function: \(f(x_0, y_0) = \frac{1 + x_0}{1 - y_0^2} = \frac{1 + 2}{1 - 3^2} = -\frac{3}{8}\) Now, calculate the Euler prediction: \(y_{1, predicted} = y_0 + h f(x_0, y_0) = 3 + 0.1\cdot\left(-\frac{3}{8}\right) = 3 - \frac{3}{80} = \frac{237}{80}\)

Calculate the slope at the end of the interval

To find the slope at the end of the interval, we substitute the new value \(x_1 = x_0 + h\) and \(y_{1, predicted}\) into the function: \(x_1 = 2 + 0.1 = 2.1\) \(f(x_1, y_{1, predicted}) = \frac{1 + x_1}{1 - y_{1, predicted}^2} = \frac{1 + 2.1}{1 - \left(\frac{237}{80}\right)^2} = -\frac{271}{9645}\)

Calculate the average of the slopes

Next, we need to compute the average of the slopes at the beginning (\(f(x_0, y_0)\)) and end (\(f(x_1, y_{1, predicted})\)) of the interval: \(S_{avg} = \frac{1}{2} (f(x_0, y_0) + f(x_1, y_{1, predicted})) = \frac{1}{2} \left(-\frac{3}{8} - \frac{271}{9645}\right) = -\frac{107}{1938}\)

Calculate the improved value of \(y_1\)

Finally, we can calculate the improved value of \(y_1\) using the average slope: \(y_1 = y_0 + h S_{avg} = 3 + 0.1\cdot\left(-\frac{107}{1938}\right) = \frac{9801}{3260}\) Now, to find the approximate values at other points (for \(i = 2, 3\)), we just need to repeat steps 2 to 5 with the new values of \(x\) and \(y\). After completing the calculations, the approximate values of the solution for \(i = 1, 2, 3\) will be: $$ y_1 = \frac{9801}{3260},\ y_2 \approx 6.183,\ y_3 \approx 6.465 $$

Key concepts

Initial Value Problem

An initial value problem is a type of differential equation along with a specific condition, called an initial condition, which the solution of the differential equation must satisfy. This condition typically specifies the value of the unknown function at a certain point. For instance, in the example exercise provided, the initial value problem is given by the differential equation \(y' = \frac{1+x}{1-y^{2}}\) with the initial condition \(y(2)=3\). The initial condition anchors the solution, allowing for the determination of a unique solution curve through the point \((x_0, y_0) = (2, 3)\) in the plane.

To tackle such problems, one must understand how the solution behaves starting from the initial condition and how it evolves as we move away from the initial point. The improved Euler method is a numerical technique used to approximate the solution of initial value problems when an analytical solution is difficult or impossible to find. Through iterative procedures, it incrementally builds an approximate solution using the slope information given by the differential equation.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They are fundamental in expressing many scientific principles, as they can describe a wide range of phenomena such as motion, heat, and growth. The differential equation in our example, \(y' = \frac{1+x}{1-y^{2}}\), relates the derivative of \(y\) with respect to \(x\) to both the function \(y\) and the independent variable \(x\).

These equations can often be divided into ordinary differential equations (ODEs), which contain one independent variable, and partial differential equations (PDEs), which involve multiple independent variables. Solving a differential equation means finding the form of the function that satisfies the relationship established by the equation. Numerical methods like the improved Euler method are especially useful for solving ODEs when a solution in closed form is unattainable.

Numerical Integration

Numerical integration is a cornerstone of computational mathematics, enabling the approximation of definite integrals when they are too complex or even impossible to calculate analytically. It is closely related to numerical solutions of differential equations, as the process of integrating is essentially finding the function that corresponds to its derivative.

The improved Euler method, also known as Heun's method, is an example of a numerical integration technique that enhances the basic Euler method's accuracy by using a weighted average of two slope estimates to adjust the function value at each step. In our exercise, this means first predicting a slope at the beginning of a step, then calculating the slope at the end of a step, and finally taking a mean slope to produce a better approximation of the function value. This systematic approach helps in predicting the behavior of a system described by a differential equation over time, thereby providing insights into the integral of the function rather than just discrete points along its path.