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Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 9 - First-Order Differential EquationsProblema 9.1.10
Capitolo 9, Problema 9.1.10

Integral Equations
In Exercises 7–12, write an equivalent first-order differential equation
and initial condition for y.


y = 1 + ∫₀ ͯ y(t) dt

Guida verificata passo dopo passo
1
Identify the given integral equation: \(y(x) = 1 + \int_0^x y(t) \, dt\).
Differentiate both sides of the equation with respect to \(x\) to eliminate the integral. Use the Fundamental Theorem of Calculus, which states that \(\frac{d}{dx} \int_0^x y(t) \, dt = y(x)\).
After differentiation, the equation becomes \(\frac{dy}{dx} = y(x)\).
Rewrite the equation as a first-order differential equation: \(y' = y\).
Determine the initial condition by evaluating the original equation at \(x=0\): \(y(0) = 1 + \int_0^0 y(t) \, dt = 1 + 0 = 1\). So, the initial condition is \(y(0) = 1\).

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Integral Equations

An integral equation expresses a function in terms of an integral that involves the function itself. Solving such equations often involves converting them into differential equations, which can be easier to analyze and solve using standard calculus techniques.
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Parameterizing Equations

Differentiation Under the Integral Sign

This technique involves differentiating an integral whose limits or integrand depend on a variable. It allows transforming integral equations into differential equations by applying the Fundamental Theorem of Calculus to differentiate both sides with respect to the variable.
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Finding Differentials

Initial Conditions for Differential Equations

Initial conditions specify the value of the unknown function at a particular point, ensuring a unique solution to a differential equation. When converting integral equations to differential form, identifying the correct initial condition is essential for solving the resulting equation.
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Solutions to Basic Differential Equations
Pratica correlata
Domanda del libro di testo

Integral Equations

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

and initial condition for y.


y = ∫₁ ͯ 1/t dt

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Domanda del libro di testo

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


y' + (tanx)y = cos²x, -π/2 < x < π/2

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Domanda del libro di testo

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


(1+x)y' + y = √x

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Domanda del libro di testo

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


(t-1)³ ds/dt + 4(t-1)²s = t+1, t >1

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Domanda del libro di testo

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' = 2xy + 2y, y(0) = 3, dx = 0.2

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Domanda del libro di testo

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.


(x+1) dy/dx - 2 (x² + x)y = exp(x²) / (x+1), x > -1, y(0) = 5

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