Ch. 9 - First-Order Differential Equations

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 9 - First-Order Differential Equations

Problem 9.1.26

Chapter 9, Problem 9.1.26

What integral equation is equivalent to the initial value problem y' = f(x), y(x₀) = y₀?

Verified step by step guidance

Start with the given initial value problem (IVP): \(y' = f(x)\) with the initial condition \(y(x_0) = y_0\).

Recall that \(y' = \frac{dy}{dx}\) represents the derivative of \(y\) with respect to \(x\).

To find an equivalent integral equation, integrate both sides of the differential equation from \(x_0\) to \(x\): \(\int_{x_0}^{x} y'(t) \, dt = \int_{x_0}^{x} f(t) \, dt\).

By the Fundamental Theorem of Calculus, the left integral simplifies to \(y(x) - y(x_0)\), so we have \(y(x) - y_0 = \int_{x_0}^{x} f(t) \, dt\).

Rearranging, the integral equation equivalent to the IVP is \(y(x) = y_0 + \int_{x_0}^{x} f(t) \, dt\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Initial Value Problem (IVP)

An initial value problem specifies a differential equation along with a value of the unknown function at a particular point. For y' = f(x), y(x₀) = y₀, the solution must satisfy both the differential equation and the initial condition y(x₀) = y₀.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, stating that if a function is continuous, its integral can be used to recover the original function up to a constant. It allows rewriting the differential equation y' = f(x) as an integral equation involving the integral of f.

Integral Equation Equivalent to IVP

The initial value problem y' = f(x), y(x₀) = y₀ can be expressed as y(x) = y₀ + ∫ₓ₀ˣ f(t) dt. This integral equation states that the solution y(x) equals the initial value plus the accumulated area under f from x₀ to x.

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Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

xdy/dx + y = e ͯ, x > 0

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Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

dy/dt + 2y = 3, y(0) = 1

Textbook Question

Solve the Bernoulli equations in Exercises 29–32.

y' - y = xy²

Textbook Question

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' = 2y/x, y(1) = -1, dx = 0.5

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Textbook Question

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

dy/dx + xy = x, y(0) = -6

Textbook Question

Using Euler’s Method

y' = y²(1+2x), (y-1) = 1, dx = 0.5