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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 9.2.15
Capitolo 9, Problema 9.2.15

Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.
dy/dt + 2y = 3, y(0) = 1

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Identify the type of differential equation. The given equation is a first-order linear ordinary differential equation of the form \(\frac{dy}{dt} + P(t)y = Q(t)\), where \(P(t) = 2\) and \(Q(t) = 3\).
Find the integrating factor \(\mu(t)\) using the formula \(\mu(t) = e^{\int P(t)\,dt}\). In this case, calculate \(\mu(t) = e^{\int 2\,dt}\).
Multiply both sides of the differential equation by the integrating factor \(\mu(t)\) to rewrite the left side as the derivative of a product: \(\frac{d}{dt}[\mu(t) y] = \mu(t) Q(t)\).
Integrate both sides with respect to \(t\) to find \(\mu(t) y = \int \mu(t) Q(t)\, dt + C\), where \(C\) is the constant of integration.
Use the initial condition \(y(0) = 1\) to solve for the constant \(C\), then solve for \(y(t)\) by dividing both sides by \(\mu(t)\).

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Concetti chiave

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First-Order Linear Differential Equations

These are differential equations of the form dy/dt + P(t)y = Q(t), where the solution involves finding an integrating factor to simplify the equation. Recognizing this form allows the use of systematic methods to solve for y(t).
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Classifying Differential Equations

Integrating Factor Method

This technique involves multiplying the entire differential equation by an integrating factor, usually e^(∫P(t)dt), to rewrite the left side as a derivative of a product. This simplifies solving the equation by enabling direct integration.
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Euler's Method

Initial Value Problems (IVP)

An IVP specifies the value of the unknown function at a particular point, such as y(0) = 1. This condition is used to find the unique solution to the differential equation that satisfies the initial condition.
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Percorso guidato
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Initial Value Problems
Pratica correlata
Domanda del libro di testo

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.


t dy/dt + 2y = t³, t > 0, y(2) = 1

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Use Euler’s method with dx = 0.5 to estimate y(5) if y′ = y²/√x and y(1) = −1. What is the exact value of y(5)?

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Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.


θ dy/dθ + y = sin θ, θ > 0, y(π/2) = 1

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Solve the Bernoulli equations in Exercises 29–32.


y' - y = xy²

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

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

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