Question

Exercises 23-27: A computer or programmable calculator is needed for these exercises. For the given initial value problem, use the Runge-Kutta method \((9)\) with a step size of \(h=0.1\) to obtain a numerical solution on the specified interval. \(y^{\prime}=-y+t, \quad y(1)=0, \quad 1 \leq t \leq 5\)

Short answer

**Answer:** The general formula to update the values of \(y_n\) is given by: $$y_{n+1}=y_n+\frac{1}{6}(k_1+2k_2+2k_3+k_4)$$ where \(k_1\), \(k_2\), \(k_3\), and \(k_4\) are computed using the given formulas.

Step-by-step solution

Define the differential equation and Runge-Kutta method formulas

We are given the differential equation of the form: $$y'(t) = f(t, y(t)) = -y + t$$ The Runge-Kutta \((9)\) method of order 4 with step size \(h = 0.1\) consists of recursively computing: $$y_{n+1}=y_n+\frac{1}{6}(k_1+2k_2+2k_3+k_4)$$ with $$k_1=0.1f(t_n,y_n)$$ $$k_2=0.1f(t_n+0.05,y_n+\frac{k_1}{2})$$ $$k_3=0.1f(t_n+0.05,y_n+\frac{k_2}{2})$$ $$k_4=0.1f(t_n+0.1,y_n+k_3)$$ and initial values \(t_0 = 1\) and \(y_0 = 0\).

Define the function f(t, y)

Our given function is \(f(t, y) = -y + t\). So, for any \((t, y)\), we can calculate the function as: $$f(t, y) = -y + t$$

Implement the Runge-Kutta method to find the numerical solution

We will use the Runge-Kutta method formulas to approximate the solution in range \(1\leq t \leq 5\) with step size \(h = 0.1\). We will iterate until \(t=5\) is reached: 1. Start with \(t_0 = 1\), \(y_0 = 0\), and \(h = 0.1\). 2. For each iteration, compute \(k_1\), \(k_2\), \(k_3\), and \(k_4\) using their formulas. 3. Compute \(y_{n+1}\) using the Runge-Kutta formula. 4. Update the values \(t_{n+1}=t_n+0.1\), \(y_n = y_{n+1}\), and repeat steps 2-4. 5. Stop iteration when \(t = 5\) is reached. With these steps, the problem can be solved using a computer or programmable calculator to obtain the numerical solution.

Key concepts

Numerical Solution

The concept of a numerical solution comes into play when we are faced with differential equations that are challenging or even impossible to solve analytically. Instead of finding an exact solution, we aim to approximate the solution using numerical methods.

Numerical solutions are particularly useful in engineering and scientific applications where exact solutions might not be feasible due to the complexity of the equations involved.

The Runge-Kutta method is a powerful tool used here, allowing us to compute an approximate solution over a given interval by making successive calculations at discrete points. This approach is iterative, meaning it builds the solution step-by-step, using previous computations to estimate future values.

By adopting this method, we can achieve a high level of accuracy, governed by the chosen step size.

Initial Value Problem

An initial value problem is a type of differential equation that requires us to determine a function which satisfies the differential equation and also fulfills a specific initial condition.

In the given problem, the initial value is specified as a point \( t_0, y_0 \) where \( t_0 = 1 \) and \( y_0 = 0 \). This starter point serves as the anchor from which all calculations proceed.

Understanding the initial value is crucial because it provides the starting conditions for the differential equation, enabling us to uniquely determine the solution that evolves from this point. In practical terms, it anchors the trajectory or the curve traced by the solution across the defined interval, ensuring consistency across the computation process.

Differential Equation

Differential equations are mathematical expressions that describe the relationship between a function and its derivatives. Essentially, they model how a system evolves over time or space.

In this exercise, the differential equation given is \(y' = -y + t\). This represents a dynamic relationship where the rate of change of the function \(y(t)\) depends linearly on the function itself and the variable \(t\).

Solving a differential equation means finding the function \(y(t)\). In many cases, as with the Runge-Kutta method, we aim to approximate the function numerically rather than finding an exact analytical expression. This particular equation is first-order, meaning it involves only the first derivative of the function. Such equations commonly arise in real-world situations from physics to finance, requiring solutions to predict behaviors over time.

Step Size

The step size, denoted by \(h\), is a critical component in numerical methods like the Runge-Kutta technique. It defines the increments in which the independent variable, typically time \(t\), progresses during the computation of the solution.

In the given problem, a small step size, \(h = 0.1\), is used. This means the algorithm computes the approximate solution at every 0.1 unit interval of \(t\).

Having a smaller step size generally increases the accuracy of the solution, as it allows for a finer granularity of approximation. However, it also requires more computational steps, leading to greater computational effort and time. Conversely, a larger step size reduces computation time but can compromise the accuracy.

• Therefore, choosing an appropriate step size is about balancing accuracy and computational efficiency.

By selecting \(h = 0.1\), a compromise is struck that aims to ensure precision while maintaining manageable computational demands.