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Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 9 - First-Order Differential Equations문제 9.2.15
9장, 문제 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 \(\mu(t)\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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).
추천 영상:
07:39
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.
추천 영상:
07:33
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.
추천 영상:
05:03
Initial Value Problems
관련 실천
교과서 질문

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


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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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교과서 질문

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


xdy/dx + y = e ͯ, x > 0

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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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교과서 질문

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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