Ch. 9 - First-Order Differential Equations

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 9 - First-Order Differential Equations

Problema 9.1.21

Capitolo 9, Problema 9.1.21

Use Euler’s method with dx = 0.2 to estimate y(1) if y′ = y and y(0) = 1. What is the exact value of y(1)?

Guida verificata passo dopo passo

Identify the differential equation and initial condition: \(\frac{dy}{dx} = y\) with \(y(0) = 1\).

Set the step size \(\Delta x = 0.2\) and note that you want to estimate \(y(1)\), so you will perform steps from \(x=0\) to \(x=1\) in increments of 0.2.

Apply Euler's method formula iteratively: \(y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n)\), where \(f(x, y) = y\) in this case.

Calculate each successive \(y\) value starting from \(y_0 = 1\) at \(x_0 = 0\), then \(y_1\) at \(x_1 = 0.2\), \(y_2\) at \(x_2 = 0.4\), and so on until \(x_5 = 1\).

For the exact value of \(y(1)\), recognize that the solution to \(\frac{dy}{dx} = y\) with \(y(0) = 1\) is \(y = e^x\), so \(y(1) = e^1 = e\).

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Euler's Method

Euler's method is a numerical technique to approximate solutions of first-order differential equations. It uses a step size (dx) to incrementally estimate the value of the function by moving along the slope given by the derivative. Starting from an initial condition, each new value is found by adding the product of the derivative and step size to the previous value.

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Euler's Method

Initial Value Problem (IVP)

An initial value problem specifies a differential equation along with a starting point (initial condition) for the function. Here, y' = y with y(0) = 1 means the solution curve passes through (0,1). The initial condition is essential for determining the unique solution to the differential equation.

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Initial Value Problems

Exact Solution of y' = y

The differential equation y' = y has the exact solution y = Ce^x, where C is a constant determined by the initial condition. Given y(0) = 1, the solution is y = e^x. Thus, the exact value of y(1) is e, approximately 2.718, which can be used to compare with the Euler's method estimate.

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Disk Method Using y-Axis

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Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.

y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt

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In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.

y' = 2xexp(x²) , y(0) = 2, dx = 0.1, x* = 1

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Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = x(1-y), y(1) = 0, dx = 0.2

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Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.

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In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.

y' = 2y²(x-1), y(2) = -1/2, dx = 0.1, x* = 3

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Show that (0, 0) and (c/d, a/b) are equilibrium points. Explain the meaning of each of these points.

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