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 \(i=1,2,3\). $$ y^{\prime}=2 x^{2}+3 y^{2}-2, \quad y(2)=1 ; \quad h=0.05 $$

Short answer

Question: Calculate the approximate solutions of the given initial value problem \(y'=2x^2+3y^2-2\) with \(y(2)=1\) and step size \(h=0.05\) at the points \(x_1 = 2.05\), \(x_2 = 2.10\), and \(x_3 = 2.15\) using the improved Euler method. Answer: The approximate solutions are \(y_1 = 1.2823125\), \(y_2 = 1.58287076\), and \(y_3 = 1.91666198\).

Step-by-step solution

Setup Improved Euler's Method equation

For improved Euler's method, we have the following equation to calculate the next value of y (y_{i+1}) using the current value of y (y_i) and x (x_i): $$ y_{i+1} = y_i + h \cdot [\frac{f(x_i,y_i) + f(x_{i+1},y^{\ast}) }{2}] $$ where, \(f(x,y) = 2x^2+3y^2-2\), the given differential equation, \(y^{\ast} = y_i + h \cdot f(x_i, y_i)\), an intermediate step, \(h = 0.05\), the given step size.

Define the iterations

Perform the three iterations mentioned in the problem (i=1,2,3). Keep track of x and y at each iteration, and find the value for the intermediate step \(y^{\ast}\) using the given function \(f(x,y)\).

First Iteration (i=1)

For i=1, the initial values are: \(x_0 = 2\), \(y_0 = 1\) 1. Calculate \(y^{\ast}\): $$ y^{\ast} = y_0 + h \cdot f(x_0, y_0) = 1 + 0.05 (2 (2)^2 + 3 (1)^2 -2) = 1.25 $$ 2. Calculate \(y_1\): $$ y_1 = y_0 + h \cdot [\frac{f(x_0,y_0) + f(x_1, y^{\ast}) }{2}] = 1 + 0.05 \cdot [\frac{(2 (2)^2 + 3 (1)^2 -2) + (2 (2.05)^2 + 3 (1.25)^2 -2)}{2}] = 1.2823125 $$ Thus, after the first iteration, \(x_1 = 2.05\) and \(y_1 = 1.2823125\).

Second Iteration (i=2)

For i=2, use the obtained values from the first iteration: \(x_1 = 2.05\), \(y_1 = 1.2823125\) 1. Calculate \(y^{\ast}\): $$ y^{\ast} = y_1 + h \cdot f(x_1, y_1) = 1.2823125 + 0.05 (2 (2.05)^2 + 3 (1.2823125)^2 -2) = 1.30396406 $$ 2. Calculate \(y_2\): $$ y_2 = y_1 + h \cdot [\frac{f(x_1,y_1) + f(x_2, y^{\ast}) }{2}] = 1.2823125 + 0.05 \cdot [\frac{(2 (2.05)^2 + 3 (1.2823125)^2 -2) + (2 (2.10)^2 + 3 (1.30396406)^2 -2)}{2}] = 1.58287076 $$ Thus, after the second iteration, \(x_2 = 2.10\) and \(y_2 = 1.58287076\).

Third Iteration (i=3)

For i=3, use the obtained values from the second iteration: \(x_2 = 2.10\), \(y_2 = 1.58287076\) 1. Calculate \(y^{\ast}\): $$ y^{\ast} = y_2 + h \cdot f(x_2, y_2) = 1.58287076 + 0.05 (2 (2.10)^2 + 3 (1.58287076)^2 -2) = 1.65939799 $$ 2. Calculate \(y_3\): $$ y_3 = y_2 + h \cdot [\frac{f(x_2,y_2) + f(x_3, y^{\ast}) }{2}] = 1.58287076 + 0.05 \cdot [\frac{(2 (2.10)^2 + 3 (1.58287076)^2 -2) + (2 (2.15)^2 + 3 (1.65939799)^2 -2)}{2}] = 1.91666198 $$ Thus, after the third iteration, \(x_3 = 2.15\) and \(y_3 = 1.91666198\). The approximate solutions of the given initial value problem at the points \(x_1 = 2.05\), \(x_2 = 2.10\), and \(x_3 = 2.15\) are \(y_1 = 1.2823125\), \(y_2 = 1.58287076\), and \(y_3 = 1.91666198\), respectively, using the improved Euler method.

Key concepts

Initial Value Problem

An initial value problem in mathematics is a type of problem where the goal is to find a function that satisfies a given differential equation and meets specified requirements, typically the value of the function and perhaps its derivatives at a given point. For instance,

In our exercise, we're given the initial value problem comprising a differential equation \( y'= 2x^2 + 3y^2 - 2 \) and an initial condition \( y(2) = 1 \). The challenge here is to determine the values of the function \( y \) at certain points \( x_i \), starting from the initial point \( x_0 \). This provides us with a foundation from which the solution can be built iteratively using numerical methods, such as the Improved Euler Method.

Understanding initial value problems is crucial in various fields like physics and engineering, where these problems often describe the evolution of some physical quantity over time or space. By solving these problems, we can predict future behavior of systems described by the differential equations.

Differential Equations

Differential equations play a pivotal role in modeling real-world phenomena where one state leads to another, continuously. A differential equation is an equation that involves one or more functions and their derivatives.

The differential equation in our current task is \( y'= 2x^2 + 3y^2 - 2 \). This particular equation is a first-order non-linear ordinary differential equation (ODE). 'First-order' indicates that only the first derivative of the unknown function, \( y' \), appears in the equation. 'Non-linear' means that the equation cannot be written as a linear combination of the unknown function and its derivatives without multiplication or division. Solving such equations analytically can be challenging or even impossible, which is why numerical methods are so valuable.

The solution to a differential equation is a function, or a set of functions, that fulfills the equation. For instance, given the initial condition \( y(2) = 1 \), we're looking for a function \( y(x) \) that satisfies the equation for each value of \( x \) when \( y \) has the specified initial value at \( x=2 \).

Numerical Methods

Numerical methods are algorithms used for computing approximations to solutions of mathematical problems that might not be feasible to solve analytically. These methods are indispensable for complex equations typical in scientific and engineering tasks.

The Improved Euler Method, also known as Heun's Method, is a numerical technique employed to provide approximate solutions to ordinary differential equations (ODEs). It's an extension of the basic Euler's Method that improves accuracy by using a sort of 'predictor-corrector' approach. Let's consider how it works using our exercise: to find subsequent values of \( y \), it averages the slope at the current point and the slope at a predicted next point. This average slope is then used to calculate the actual next value of \( y \).

Algorithm steps:

1. First predict \( y^* \) by taking a step using the current slope.
2. Find an estimate of the slope at the predicted next point.
3. Take the average of these two slopes.
4. Use the average slope to determine the next value of \( y \).

This method provides a balance between computational efficiency and accuracy, making it a suitable choice for many practical applications. By applying the Improved Euler Method to the initial value problem provided, one can find approximations to the solution at specified points, offering insight into the behavior of systems modeled by the ODE.