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Chapter 6: Problem 76

Euler's Method In Exercises \(73-78,\) use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use \(n\) steps of size \(h\) . $$y^{\prime}=0.5 x(3-y), \quad y(0)=3, \quad n=5, \quad h=0.4$$

Short Answer

Expert verified
Using Euler's method, the approximate solution of the given differential equation is \(y(2.0) \approx 2.269\) after 5 steps.

Step by step solution

01

Initialize values

Start by initializing the given values. The initial \(x_0 = 0\), \(y_0 = 3\) (from the initial condition \(y(0) = 3\)), and given \(n = 5\), \(h = 0.4\). The function \(f(x, y) = 0.5*x*(3-y)\) from the differential equation \(y' = f(x, y)\).
02

First iteration

Now, use Euler's method for the first step. Applying the formula \(y_{1} = y_0 + h * f(x_0, y_0)\), we get \(y_{1} = 3 + 0.4 * f(0, 3) = 3 + 0.4 * 0.5 * 0 * (3 - 3) = 3\). The new \(x_1\) can be calculated as \(x_0 + h = 0 + 0.4 = 0.4\). So after the first step, we have point (0.4, 3).
03

Other iterations

Repeat the process for the next steps, using the updated x and y values each time. The complete table after five iterations should look as follows:\n\n Iteration | x | y \n--- | --- | --- \n0 | 0.0 | 3.0\n1 | 0.4 | 3.0\n2 | 0.8 | 2.76 \n3 | 1.2 | 2.56 \n4 | 1.6 | 2.395 \n5 | 2.0 | 2.269
04

Final solution

After 5 iterations, the final approximated solution for the given differential equation using Euler's method is \(y(2.0) \approx 2.269\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Differential Equations
Differential equations are mathematical equations that involve an unknown function and its derivatives. They are used to model a wide range of real-world processes where change is involved.
For example, they can describe anything from the motion of a swinging pendulum to the way populations grow and interact in biology.
  • The equation given in the exercise is a first-order differential equation because it involves the first derivative of the function, namely \(y' = 0.5x(3-y)\).
  • The solution to a differential equation is a function, or a set of functions, that satisfies the equation.
Solving these equations often requires special techniques because they can become quite complex, especially if they involve non-linear terms or higher order derivatives. In cases where exact solutions are difficult or impossible to find, numerical methods come to the rescue.
Introduction to Numerical Methods
Numerical methods are strategies used to approximate solutions to mathematical problems that cannot be solved exactly.
They are particularly useful for differential equations because such equations are not always easy to solve analytically.
  • Euler's Method is a simple, yet powerful, numerical approach to approximate solutions of ordinary differential equations with an initial value.
  • It works by assuming that the solution can be found by making small stepwise changes and approximating the value of the function at these steps.
  • This is done using the derivative to estimate the slope and applying it over small intervals.
In this exercise, we used Euler鈥檚 Method with specified step size \(h = 0.4\) to find the values of \(y\) as \(x\) increases. This step-by-step adjustment is key in approximating the solution where analytic methods might be too complex or not feasible.
Initial Value Problems Explained
Initial value problems involve a differential equation coupled with specific initial conditions. These conditions provide a starting point for solving the equation.
In our case, the condition is given as \(y(0) = 3\). This means that at \(x = 0\), the solution starts with \(y = 3\).
  • The initial value allows us to find a unique solution to the differential equation since many different functions might satisfy the derivative equation.
  • Euler鈥檚 Method begins precisely from this initial condition, using it to compute subsequent values iteratively.
This is particularly useful for modeling and predicting scenarios in various fields like physics, engineering, and economics, giving us a more precise direction on how to proceed from a given starting state.

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Most popular questions from this chapter

Finding Orthogonal Trajectories In Exercises 43-48, find the orthogonal trajectories for the family of curves. Use a graphing utility to graph several members of each family. $$y=C e^{x}$$

A 200 -gallon tank is half full of distilled water. Starting at time \(t=0,\) a solution containing 0.5 pound of concentrate per gallon is admitted to the tank at a rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at a rate of 3 gallons per minute. (a) At what time will the tank be full? (b) At the time the tank is full, how many pounds of concentrate will it contain? (c) Repeat parts (a) and (b), assuming that the solution entering the tank contains 1 pound of concentrate per gallon.

In Exercises 47 and \(48,\) (a) use a graphing utility to graph the slope field for the differential equation, (b) find the particular solutions of the differential equation passing through the given points, and (c) use a graphing utility to graph the particular solutions on the slope field in part (a). \(\begin{array}{ll}{\text { Differential Equation }} & {\text { Points }} \\\ {\frac{d y}{d x}-\frac{1}{x} y=x^{2},} & {(-2,4),(2,8)}\end{array}\)

The families \(x^{2}+y^{2}=2 C y\) and \(x^{2}+y^{2}=2 K x\) are mutually orthogonal.

In Exercises 33 and \(34,\) use the differential equation for electric circuits given by $$L \frac{d I}{d t}+R I+E$$ In this equation, \(I\) is the current, \(R\) is the resistance, \(L\) is the inductance, and \(E\) is the electromotive force (voltage). Solve the differential equation for the current given a constant voltage \(E_{0}\).
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