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Chapter 6: Problem 54

Percentage Error Let \(y=f(x)\) be solution to the initial value problem \(d y / d x=2 x-1\) such that \(f(2)=3 .\) Find the per- centage error if Euler's Method with \(\Delta x=-0.1\) is used to ap- proximate \(f(1.6) .\)

Short Answer

Expert verified
Percentage error is calculated using the values obtained from the exact equation and the Euler's method, then using these to find the difference between the exact and approximate values relative to the exact value, multiplied by 100 to put it in terms of percentage. Using mathematical calculations, we can solve this value numerically.

Step by step solution

01

Solve The Differential Equation

Given the differential equation \(dy/dx = 2x - 1\) with the initial condition \(f(2)=3\), we can find its general solution by integrating both sides. After Integrating, the solution to the equation becomes \(f(x) = x^2 - x + c\). We can substitute \(x = 2\) and \(f(2) = 3\) into the equation to find \(c = -1\). Thus the exact solution for the differential equation is \(f(x) = x^2 - x - 1\).
02

Euler's Method Implementation

Euler's method is a numerical method used to approximate solutions of first order differential equations. It is defined by the recurrence relation \(y_{n+1} = y_n +hf(x_n, y_n)\), where in our case, \(h=-0.1\) and \(f(x, y) = 2x- 1\). Starting with \(x_0=2\), \(y_0=f(2)=3\) we apply the recurrence relation five times to reach \(x=1.6\).
03

Calculate The Percentage Error

The percentage error is calculated using the formula \((\text{exact value} - \text{approximate value}) / \text{exact value} * 100\%\). In this case, the exact value is given by \(f(1.6) = (1.6)^2 - 1.6 -1\) and the approximate value is given by the result at \(x=1.6\) from Euler's method.

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Key Concepts

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

Initial Value Problem
In mathematics, an initial value problem refers to a type of differential equation that comes with a specific condition. This condition is known as the 'initial condition.' The problem starts by setting the value of the function, which also needs to satisfy the differential equation at a certain point.

For instance, in the exercise above, the differential equation is \(dy/dx = 2x - 1\), and the initial condition is given as \(f(2)=3\). This means that when \(x=2\), the function \(y\) must equal 3.

To solve the problem, you integrate the differential equation to find a general solution, which usually includes an arbitrary constant \(c\). Then, you use the initial condition to determine this constant, ensuring that the solution satisfies the given values. Solving such problems is crucial as it provides a precise function that can predict the behavior of a system from a known starting point.
Differential Equation
Differential equations are mathematical expressions used to describe the relationship between a function and its derivatives. These equations are essential in various scientific and engineering fields because they model how quantities change over time or space.

The differential equation given in the exercise is \(dy/dx = 2x - 1\). It indicates how the derivative of \(y\) concerning \(x\) behaves. Solving a differential equation involves finding a function that fits this behavior, which can often demonstrate natural phenomena like motion, growth, or decay.

Understanding and working with differential equations is central to solving real-life problems, where exact solutions help make predictions and inform decisions. In our specific problem, solving the equation means determining the function \(f(x)\) that describes how \(y\) changes with \(x\), which is a key step in calculating both exact outcomes and understanding errors in numerical approximations.
Numerical Approximation
Numerical approximation is a method used when finding an exact solution to a problem is either impossible or impractical. Approximations come in handy to provide insight into the behavior of systems modeled by differential equations.

Euler's Method is a simple yet powerful numerical technique for approximating solutions. It uses a step-by-step approach to estimate the values of the function at discrete points. Here, Euler's method involves starting from an initial point \((x_0, y_0)\), using the recurrence relation \(y_{n+1} = y_n + h \cdot f(x_n, y_n)\), where \(h\) is the step size.

For instance, in the exercise, we use a negative step size \(\Delta x=-0.1\) to approximate \(f(1.6)\) from \(f(2)=3\). The method progresses through a series of calculated steps, providing an approximate value that can be contrasted with the exact solution to understand its accuracy.
  • Euler's Method is straightforward but can accumulate errors over longer ranges or larger step sizes.
  • The smaller the step size, the closer the approximation to the exact solution.
  • It's practiced when analytical methods are complex or impossible to apply.
Analyzing percentage error helps assess the reliability of this approximation technique, highlighting both its strengths and limitations.

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Most popular questions from this chapter

You should solve the following problems without using a graphing calculator. True or False For small values of \(t\) the solution to logistic differential equation \(d P / d t=k P(100-P)\) that passes through the point \((0,10)\) resembles the solution to the differential equa- tion \(d P / d t=k P\) that passes through the point \((0,10) .\) Justify your answer.

In Exercises \(21-24\) , solve the initial value problem using the Fundamental Theorem. (Your answer will contain a definite integral.) \(G^{\prime}(s)=\sqrt[3]{\tan s}\) and \(G(0)=4\)

Finding the Original Temperature of a Beam An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at \(65^{\circ} \mathrm{F}\) . After 10 min, the beam warmed to \(35^{\circ} \mathrm{F}\) and after another 10 \(\mathrm{min}\) its temperature was \(50^{\circ} \mathrm{F} .\) Use Newton's Law of Cooling to estimate the beam's initial temperature.

Skydiving If a body of mass \(m\) falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity \(v(t)\) is modeled by the initial value problem $$\begin{array}{ll}{\text { Differential equation: }} & {m \frac{d v}{d t}=m g-k v^{2}} \\ {\text { Initial condition: }} & {v(0)=0}\end{array} $$ where \(t\) represents time in seconds, \(g\) is the acceleration due to gravity, and \(k\) is a constant that depends on the body's aerodynamic properties and the density of the air. (We assume that the fall is short enough so that variation in the air's density will not affect the outcome.) (a) Show that the function $$v(t)=\sqrt{\frac{m g}{k}} \frac{e^{a t}-e^{-a t}}{e^{a t}+e^{-a t}}$$ where \(a=\sqrt{g k / m},\) is a solution of the initial value problem. (b) Find the body's limiting velocity, \(\lim _{t \rightarrow \infty} v(t)\) (c) For a 160 -lb skydiver \((m g=160),\) and with time in seconds and distance in feet, a typical value for \(k\) is \(0.005 .\) What is the diver's limiting velocity in feet per second? in miles per hour?

Resistance Proportional to Velocity It is reasonable to assume that the air resistance encountered by a moving object, such as a car coasting to a stop, is proportional to the object's velocity. The resisting force on an object of mass \(m\) moving with velocity \(v\) is thus \(-k v\) for some positive constant \(k\). (a) Use the law Force = Mass \(\times\) Acceleration to show that the velocity of an object slowed by air resistance (and no other forces) satisfies the differential equation $$m \frac{d y}{d t}=-k v$$ (b) Solve the differential equation to show that \(v=v_{0} e^{-(k / m) t}\) where \(v_{0}\) is the velocity of the object at time \(t=0 . (c) If \)k$ is the same for two objects of different masses, which one will slow to half its starting velocity in the shortest time? Justify your answer.
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