Question

Estimate the following solutions using Euler's method with \(n=5\) steps over the interval \(t=[0,1] .\) If you are able to solve the initialvalue problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler's method. How accurate is Euler's method? $$ y^{\prime}=3 t-y, y(0)=1 \text { . Exact solution is } y=3 t+4 e^{-t}-3 $$

Short answer

Euler's method is approximately 0.13144 off from the exact solution at \( t = 1 \).

Step-by-step solution

Set Up the Initial Conditions

The given differential equation is \( y' = 3t - y \) with the initial condition \( y(0) = 1 \). We will apply Euler's method over the interval \( t = [0, 1] \) with \( n = 5 \) steps. This means the step size, \( h \), is \( h = \frac{1 - 0}{5} = 0.2 \).

Apply Euler's Formula Iteratively

Euler's method formula is \( y_{n+1} = y_n + h \, f(t_n, y_n) \) where \( f(t, y) = 3t - y \). Calculate iteratively starting from \( t_0 = 0 \), \( y_0 = 1 \).- **Iteration 1:** - \( t_1 = 0 + 0.2 = 0.2 \) - \( y_1 = 1 + 0.2(3(0) - 1) = 1 - 0.2 = 0.8 \)- **Iteration 2:** - \( t_2 = 0.2 + 0.2 = 0.4 \) - \( y_2 = 0.8 + 0.2(3(0.2) - 0.8) = 0.8 + 0.2(0.6 - 0.8) = 0.76 \)- **Iteration 3:** - \( t_3 = 0.4 + 0.2 = 0.6 \) - \( y_3 = 0.76 + 0.2(3(0.4) - 0.76) = 0.76 + 0.2(1.2 - 0.76) = 0.832 \)- **Iteration 4:** - \( t_4 = 0.6 + 0.2 = 0.8 \) - \( y_4 = 0.832 + 0.2(3(0.6) - 0.832) = 0.832 + 0.2(1.8 - 0.832) = 1.0424 \)- **Iteration 5:** - \( t_5 = 0.8 + 0.2 = 1.0 \) - \( y_5 = 1.0424 + 0.2(3(0.8) - 1.0424) = 1.0424 + 0.2(2.4 - 1.0424) = 1.34008 \)

Calculate the Exact Solution

The exact solution is given as \( y = 3t + 4e^{-t} - 3 \). Plugging in \( t = 1 \):\[y(1) = 3(1) + 4e^{-1} - 3 = 3 + \frac{4}{e} - 3 \approx 1.47152.\]

Evaluate the Accuracy of Euler's Method

The Euler's method approximation at \( t = 1 \) is \( y_5 = 1.34008 \). The exact solution at \( t = 1 \) is approximately \( 1.47152 \). The error is:\[|1.47152 - 1.34008| \approx 0.13144.\]The error indicates that the Euler's method provides a reasonable approximation but with noticeable error compared to the exact solution.

Key concepts

Differential Equations

Differential equations are powerful mathematical tools used to describe the relationship between a function and its derivatives. They are essential in modeling how quantities change throughout processes, such as growth and decay. A differential equation like \( y' = 3t - y \) involves a derivative (\( y' \)) and a function of one or more variables.

Understanding differential equations is crucial because they help predict future behaviors in many real-world conditions, such as in physics, engineering, and economics. The task is often to solve these equations to find the unknown function, \( y(t) \), that defines a system's behavior over time.

Solving differential equations can sometimes be done analytically, yielding explicit solutions like \( y = 3t + 4e^{-t} - 3 \). However, not all differential equations are easy to solve this way, requiring alternative strategies like numerical methods when complexity rises. With tools such as Euler's method, you can approximate solutions even when traditional methods fail.

Initial Value Problems

An initial value problem (IVP) is a differential equation that includes a specified initial condition. For instance, in the equation \( y' = 3t - y \), the initial condition is given as \( y(0) = 1 \), meaning at time \( t = 0 \), \( y \) is exactly \( 1 \).

The initial condition is critical as it allows for the determination of a unique solution to the differential equation. It provides the necessary starting point from which the equation can unfold over the desired interval.

• This is akin to knowing the starting position of an object to predict its future movement based on its velocity.
• With Euler's method, the initial condition sets off the iterative process of approximation, feeding the first known value into a step-by-step calculation pattern.

The interaction between the differential equation and its initial conditions guides the path of the solution, allowing detailed modeling of dynamic systems.

Numerical Approximation Techniques

Numerical approximation techniques like Euler's method provide practical solutions to differential equations that may not have straightforward analytical answers. These methods break up the problem into smaller, more manageable steps, making complex computations feasible.

Euler's method, for instance, involves taking an initial point, in this case, \( y(0) = 1 \), and incrementally calculating successive points using the formula \( y_{n+1} = y_n + h \cdot f(t_n, y_n) \). Here, \( h \) represents the step size derived from dividing the interval into smaller sections.

• With this approach, you incrementally build an approximate solution. Each step is based on the previously computed value.
• This iterative process allows students and engineers alike to estimate outcomes effectively when precise solutions are unattainable.
• While useful, Euler's method can introduce errors, especially in cases with higher step sizes or non-linear behavior, thereby making accuracy checks essential.

These techniques play a pivotal role in fields requiring reliable predictions where exact analytical answers may not be accessible.