Question

Use Euler's method to find a numerical solution of \(\frac{\mathrm{dy}}{\mathrm{d} x}=\frac{x y}{x^{2}+2}\) subject to \(y(1)=3\). Take \(h=0.1\) and hence approximate \(y(1.5)\). Obtain the true solution using the method of separation of variables. Work throughout to six decimal places.

Short answer

Question: Using Euler's method with step size \(h = 0.1\) and the initial condition \(y(1) = 3\), approximate the value \(y(1.5)\) for the given differential equation \(\frac{\mathrm{dy}}{\mathrm{d}x} = \frac{xy}{x^2+2}\). Compare it with the true solution obtained using the separation of variables method. Answer: Using Euler's method, we found the approximate value \(y(1.5) \approx 3.546841\). By solving the differential equation using the separation of variables method, we found the true solution to be \(y(x) = \frac{3}{\sqrt{3}}(x^2 + 2)^{\frac{1}{2}}\), and the true value of \(y(1.5)\) to be approximately \(3.535534\). The numerical approximation and the true solution are quite close, with a small error for \(y(1.5)\).

Step-by-step solution

Write down the given differential equation and initial condition

The given differential equation is \(\frac{\mathrm{dy}}{\mathrm{d}x} = \frac{xy}{x^2+2}\) with the initial condition \(y(1) = 3\).

Apply Euler's method with step size \(h = 0.1\)

Using Euler's method, we have the following iterative formula for approximating the solution: \(y_{n+1} = y_n + hf(x_n, y_n)\), where \(f(x, y) = \frac{xy}{x^2 + 2}\), \(h = 0.1\), and \(y_0 = 3\) when \(x_0 = 1\). Now, we'll find the first few points to get an idea of the solution: Iteration 1: \(x_1 = x_0 + h = 1 + 0.1 = 1.1\) \(y_1 = y_0 + h\cdot f(x_0, y_0) = 3 + 0.1(\frac{(1)(3)}{1^2+2}) = 3 + 0.1(1) = 3.1\) Iteration 2: \(x_2 = x_1 + h = 1.1 + 0.1 = 1.2\) \(y_2 = y_1 + h\cdot f(x_1, y_1) = 3.1 + 0.1(\frac{(1.1)(3.1)}{(1.1)^2+2}) \approx 3.197183\) And so on... We can continue this process until we find the value of \(y\) when \(x=1.5\).

Calculate \(y(1.5)\) using Euler's method

Iterating through this process, we find the approximate value \(y(1.5) \approx 3.546841\) (up to six decimal places).

Solve the differential equation using separation of variables

Now, we'll find the true solution using the separation of variables method. The given equation is \(\frac{\mathrm{dy}}{\mathrm{d}x} = \frac{xy}{x^2+2}\). Rearranging and separating variables, we have \(\frac{\mathrm{d}y}{y}=\frac{x\mathrm{d}x}{x^{2}+2}\). Now integrate both sides: \(\int\frac{\mathrm{d}y}{y}=\int\frac{x\mathrm{d}x}{x^{2}+2}\). Solving the integrals, we get: \(\ln |y| = \frac{1}{2}\ln|x^2+2|+C_1\). Taking exponentials of both sides, we have: \(y = e^{\frac{1}{2}\ln|x^2+2| + C_1}\) \(\Rightarrow\) \(y = e^{C_1}(x^2+2)^{\frac{1}{2}}\). Using the initial condition \(y(1) = 3\), we get: \(3 = e^{C_1}(1^2 + 2)^{\frac{1}{2}}\) \(\Rightarrow\) \(C_1=\ln\frac{3}{\sqrt{3}}\). So the true solution is \(y(x) = \frac{3}{\sqrt{3}}(x^2 + 2)^{\frac{1}{2}}\).

Calculate the true value of \(y(1.5)\)

Now, we'll find the true value of \(y(1.5)\) using the true solution: \(y(1.5) = \frac{3}{\sqrt{3}}((1.5)^2 + 2)^{\frac{1}{2}} \approx 3.535534\) (up to six decimal places). By comparing the true solution and our numerical approximation using Euler's method, we can see that they are quite close, with a small error for \(y(1.5)\).

Key concepts

Numerical Solution of Differential Equations

Numerical methods like Euler's method are invaluable when finding exact solutions to differential equations is difficult or impossible. In essence, these methods break down the complex, continuous problems into manageable, discrete steps. Euler's method is one of the simplest numerical approaches for solving first-order ordinary differential equations—those involving rates of change and dependent on one variable. To apply Euler's method, we start with an initial value and then use the equation's slope to step forward in small increments, called the step size. By iterating this process, we inch closer to the solution over the interval of interest.

However, it's important to remember Euler's method gives an approximate rather than an exact solution. The accuracy is contingent on the step size; a smaller step will generally yield a more precise approximation but will require more computations. Therefore, it's a balance between computational cost and accuracy.

Separation of Variables

The method of separation of variables is a technique used to solve certain types of ordinary differential equations. It operates under the premise of rearranging the equation so that each variable and its differential are on opposite sides of the equation. After separating the variables, we integrate both sides to find a general solution. Then, apply any given initial conditions to determine the particular solution. It's a powerful method but applicable only when the equation can be manipulated such that all terms involving one variable are isolated on one side of the equation. This technique is also a great way to validate the results obtained from numerical methods or to provide a benchmark when assessing their accuracy.

With our exercise in question, after finding the general solution through separation of variables, we used the initial condition to solve for the constant, subsequently obtaining the exact solution.

Initial Value Problem

An initial value problem in the context of differential equations is a problem where the solution to the differential equation is to be found subject to a given set of initial conditions. These conditions specify the value of the unknown function (often the position or concentration) at a particular point (often time zero). Initial value problems are critical as they ground the solution in a physical context, yielding a specific trajectory or behavior amongst a family of possible solutions. In our Euler's method exercise, the initial value is given by the condition that when the independent variable (in this case, 'x') is 1, the dependent variable ('y') is 3. This specific initial condition is what guides the numerical method to produce a single trajectory out of the infinite possibilities.

Numerical Approximation

Numerical approximation involves using computational techniques to approximate the solutions of mathematical problems that may not have straightforward analytical solutions. The idea underpinning numerical approximation is not to find the exact result but rather a value that is close enough to the correct solution within a certain tolerance. Euler's method is an example of numerical approximation where the change in the solution over each step is estimated using the derivative, which acts as the rate of change. Since it's an approximation, there's an inherent error that can be described in terms of the local truncation error at each step and the global error over the entire interval.

Improving the accuracy involves reducing the step size, but at the cost of increased computational effort. For students working with numerical approximations, understanding the trade-offs between precision, computational demand, and the impacts of cumulative error is essential.