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Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 9 - First-Order Differential EquationsProblem 9.1.10
Chapter 9, Problem 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

Verified step by step guidance
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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Key Concepts

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

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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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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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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Related Practice
Textbook Question

Integral Equations

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

and initial condition for y.


y = ∫₁ ͯ 1/t dt

Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


y' + (tanx)y = cos²x, -π/2 < x < π/2

Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


(1+x)y' + y = √x

Textbook Question

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

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

Textbook Question

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