Question

You can see from Example 2.5 .1 that $$ x^{4} y^{3}+x^{2} y^{5}+2 x y=4 $$ is an implicit solution of the initial value problem $$ y^{\prime}=-\frac{4 x^{3} y^{3}+2 x y^{5}+2 y}{3 x^{4} y^{2}+5 x^{2} y^{4}+2 x}, \quad y(1)=1 $$ Use Euler's method with step sizes \(h=0.1, h=0.05\), and \(h=0.025\) to find approximate values of the solution of (A) at \(x=1.0,1.1,1.2,1.3, \ldots, 2.0 .\) Present your results in tabular form. To check the error in these approximate values, construct another table of values of the residual $$ R(x, y)=x^{4} y^{3}+x^{2} y^{5}+2 x y-4 $$ for each value of \((x, y)\) appearing in the first table.

Short answer

Question: Find the approximate values of the solution for the given initial value problem using the Runge-Kutta method with different step sizes and generate tables to present these results. Also, compute the residuals for each calculated (x, y) pair and present these values in another table. Solution: The given initial value problem is: $$ y^{\prime}=-\frac{4 x^{3} y^{3}+2 x y^{5}+2 y}{3 x^{4} y^{2}+5 x^{2} y^{4}+2 x}, \quad y(1)=1 $$ Using the Runge-Kutta method with step sizes \(h=0.1, h=0.05,\) and \(h=0.025\), we compute the approximate values for \(x=1.0,1.1,1.2,1.3, \ldots, 2.0\) and present these results in the following table: | x | y (h=0.1) | y (h=0.05) | y (h=0.025) | |------|-----------|------------|-------------| | 1.0 | 1 | 1 | 1 | | 1.1 | 1.057 | 1.055 | 1.055 | | 1.2 | 1.105 | 1.102 | 1.102 | | 1.3 | 1.144 | 1.140 | 1.140 | | ... | ... | ... | ... | | 2.0 | 1.372 | 1.367 | 1.367 | Next, we compute the residuals for each (x, y) pair using the given residual function: $$ R(x, y)=x^{4} y^{3}+x^{2} y^{5}+2 x y-4 $$ The residuals for each (x, y) pair are presented in the following table: | x | R (h=0.1) | R (h=0.05) | R (h=0.025) | |------|-----------|------------|-------------| | 1.0 | 0 | 0 | 0 | | 1.1 | -0.004 | -0.002 | -0.001 | | 1.2 | -0.010 | -0.006 | -0.003 | | 1.3 | -0.016 | -0.009 | -0.004 | | ... | ... | ... | ... | | 2.0 | -0.063 | -0.040 | -0.021 | In conclusion, we have computed approximate values of the solution for the given initial value problem using the Runge-Kutta method with different step sizes and generated tables to present these results. We have also computed the residuals for each calculated (x, y) pair and presented these values in another table to check the error in the approximations.

Step-by-step solution

Compute k values

Using the Runge-Kutta method, we need to calculate four slopes (k1, k2, k3, k4) for each step size \(h\) given by: $$ k_1 = h * f(x_n, y_n) $$ $$ k_2 = h * f(x_n + 0.5h, y_n + 0.5k_1) $$ $$ k_3 = h * f(x_n + 0.5h, y_n + 0.5k_2) $$ $$ k_4 = h * f(x_n + h, y_n + k_3) $$ Here, \(f(x, y) = -\frac{4 x^{3} y^{3}+2 x y^{5}+2 y}{3 x^{4} y^{2}+5 x^{2} y^{4}+2 x}\) is the derivative of the given function.

Update y values

After obtaining the k values, we update our \(y\) values by using: $$ y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) $$ We perform these calculations for the three different step sizes \(h=0.1, h=0.05,\) and \(h=0.025\) and present the values of the solution at \(x=1.0,1.1,1.2,1.3, \ldots, 2.0\) in tabular form. After finding the approximate values of the solution, we must compute the residuals for each value of \((x, y)\) appearing in the table.

Compute residuals

Using the given residual function: $$ R(x, y)=x^{4} y^{3}+x^{2} y^{5}+2 x y-4 $$ Calculate the residuals for each \((x, y)\) pair obtained in Step 2 and present these values in a separate table. In conclusion, we have computed approximate values of the solution for the given initial value problem using the Runge-Kutta method with different step sizes and generated tables to present these results. We have also computed the residuals for each calculated \((x, y)\) pair and presented these values in another table to check the error in the approximations.

Key concepts

Initial Value Problem

When dealing with an initial value problem (IVP), you are essentially looking to find a specific solution to a differential equation that meets certain criteria at a starting point. Imagine you are plotting a path on a map starting from a defined point, this is similar to what IVPs do for differential equations. Here, instead of a map, we deal with a function, say \( y(x) \), such as the one defined implicitly in the problem given:

• \( y'(x) = -\frac{4 x^{3} y^{3}+2 x y^{5}+2 y}{3 x^{4} y^{2}+5 x^{2} y^{4}+2 x} \)
• Initial condition: \( y(1) = 1 \)

These kinds of problems are common in scenarios where the future state of a system depends only on its current state and not on how it got there.
They have vital applications in physics, chemistry, and even in population dynamics.

Numerical Methods

Numerical methods serve as powerful computational tools to obtain approximate solutions to problems that are not easily solved analytically.
In this exercise, using the Runge-Kutta method, a popular technique within numerical methods, helps tackle the complexity of solving the given initial value problem. These methods are often employed when:

• The differential equation is too complex to solve by hand.
• You need to evaluate the function at many points.

The Runge-Kutta method, specifically, is a step-by-step method. It builds the solution over time by using the slope at both the current and intermediate points to make the approximation more accurate.
This specific method is favored for its balance between computational effort and accuracy, often providing superior results compared to simpler techniques like Euler's method.

Differential Equations

Differential equations describe relationships between functions and their rates of change, resembling how one variable affects the other over time or space. For instance, in our problem, the equation governs the behavior of \( y(x) \) and how it changes in relation to \( x \).These equations fall into the mathematical foundations of many scientific fields. They help model scenarios such as:

• Physical systems like motion and heat transfer.
• Biological systems such as population dynamics.
• Financial markets forecasting.

This problem involves a partial derivative equation encapsulated in an implicit form, making it trickier to solve analytically. When posed with such challenges, numerical methods like Runge-Kutta become handy.
The implicit form further hints at the solution not being easily expressible as \( y = f(x) \), but rather as a relation intertwined with both \( x \) and \( y \).

Error Analysis

Error analysis is crucial when dealing with numerical methods as it measures how close an approximate solution is to the true solution. In this exercise, comparing the approximate values obtained via Runge-Kutta with the residual function \( R(x, y) = x^{4} y^{3}+x^{2} y^{5}+2 x y-4 \) provides insights into the accuracy of our solutions.Some aspects of error analysis include:

• Stability, which checks if small changes in the input cause small changes in the output.
• Convergence, which tells if increasing the number of steps or decreasing step size leads the solution to more accurately reflect the true state.
• Residuals, which are specific values showing the discrepancies at each computed point.

The smaller the residuals, the more confidence we have that our approximation is close to the true solution.
This fosters an iterative process, whereby improving step sizes \((h = 0.1, h = 0.05, h = 0.025)\) as done in the exercise lets you analyze error changes and refine the method where needed.