Question

Use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=2 y, y(0)=3,[0,1] $$

Short answer

Using Euler's method, the approximate solution at \(t = 1\) is \(y \approx 16.83456\).

Step-by-step solution

Identify Initial Condition and Parameters

The initial condition given is \(y(0) = 3\). The derivative is \(y' = 2y\). The step size \(h\) is \(0.2\), and we need to find approximate values over the interval \([0, 1]\).

Apply Euler’s Method Formula

Euler's method formula is \(y_{n+1} = y_n + h f(t_n, y_n)\), where \(f(t, y) = 2y\). Start with \(t_0 = 0\) and \(y_0 = 3\).

Calculate Successive Approximations

- For \(n = 0\): \(t_1 = t_0 + h = 0.2\), \(y_1 = y_0 + 0.2 \cdot 2y_0 = 3 + 0.2 \cdot 6 = 4.2\).- For \(n = 1\): \(t_2 = t_1 + h = 0.4\), \(y_2 = y_1 + 0.2 \cdot 2y_1 = 4.2 + 0.2 \cdot 8.4 = 5.88\).- For \(n = 2\): \(t_3 = t_2 + h = 0.6\), \(y_3 = y_2 + 0.2 \cdot 2y_2 = 5.88 + 0.2 \cdot 11.76 = 8.232\).- For \(n = 3\): \(t_4 = t_3 + h = 0.8\), \(y_4 = y_3 + 0.2 \cdot 2y_3 = 8.232 + 0.2 \cdot 16.464 = 11.5248\).- For \(n = 4\): \(t_5 = t_4 + h = 1.0\), \(y_5 = y_4 + 0.2 \cdot 2y_4 = 11.5248 + 0.2 \cdot 23.0496 = 16.83456\).

Key concepts

Numerical Approximation

Numerical approximation is a powerful mathematical technique used to estimate solutions to equations or models that cannot be solved precisely. It is extremely valuable in situations where exact solutions are difficult to obtain or do not exist.
In calculus, numerical methods help us bridge the gap between theoretical concepts and real-world applications.

• They provide a way to handle complex problems that involve intricate calculations.
• Methods like Euler's Method enable us to approximate solutions for differential equations by stepping through intervals in small increments.
• This approach is handy in a variety of fields, including physics, engineering, and economics.

Euler's Method, for instance, is the simplest form of numerical approximation for solving differential equations.
It uses the idea of linear approximation by stepping forward in small steps, using a given slope to predict the next value.
Although it might not always provide the most accurate result, its simplicity makes it a favorite starting point in learning numerical methods.

Differential Equations

Differential equations are equations that involve an unknown function and its derivatives.
They are fundamental in describing real-world phenomena, especially where change is involved over continuous time or space.

• These equations help us model various situations such as population growth, heat conduction, fluid dynamics, and much more.
• The 'differential' part refers to the change, emphasizing the relationship between differing quantities over an instance.

In the exercise provided, the differential equation is given as \(y' = 2y\).
This means that the rate of change of the function \(y\) is proportional to its current value, which is a common form of a first-order differential equation.
Solving such equations analytically can sometimes be challenging, especially in complex cases.
Fortunately, numerical methods like Euler’s Method provide a practical way to approximate solutions even when exact analytical solutions are difficult to find.

Initial Value Problems

Initial value problems (IVPs) are a type of differential equation accompanied by a specific condition, known as the initial condition.
This condition defines the starting point of the function and is essential in finding a unique solution.

• An initial value problem typically has the form \(y'(t) = f(t, y)\) with \(y(t_0) = y_0\) as the initial condition.
• The initial condition acts like a kick-off point, helping us trace the path of the solution curve from a known starting location.

The problem given in our exercise is indeed an initial value problem with the condition \(y(0) = 3\).
This sets the stage for applying Euler's Method, where the first value, \(y_0 = 3\), is used to begin the approximation process.
By following the defined steps, we calculate successive approximations for each interval, satisfying both the differential equation and the initial condition.

Calculus Education

Calculus education introduces students to different types of calculus, including differential, integral, and multivariable calculus.
These areas form the foundation of mathematical analysis and provide tools to understand change, rates, and accumulation in various contexts.

• Differential calculus deals with derivatives, which help measure how a quantity changes over time.
• Integral calculus focuses on accumulation of quantities and finding areas under curves.
• Numerical methods, such as Euler's Method, offer practical techniques embedded in calculus education to develop problem-solving skills.

Through tools like Euler's Method, students grasp the idea that some problems can have solutions that are easy to approximate but hard to solve exactly.
This fosters critical thinking as students learn to apply mathematical concepts to solve problems in both science and engineering.
Ultimately, understanding these principles equips students with the skills needed to tackle complex real-world challenges effectively.