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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 9 - Differential Equations문제 9.4.15
9장, 문제 9.4.15

11–16. Initial value problems Solve the following initial value problems.


y'(t) − 3y = 12, y(1) = 4

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1
Identify the type of differential equation: This is a first-order linear differential equation of the form \(y'(t) + p(t)y = q(t)\), where \(p(t) = -3\) and \(q(t) = 12\).
Find the integrating factor \(\mu(t)\) using the formula \(\mu(t) = e^{\int p(t) \, dt}\). Here, calculate \(\mu(t) = e^{\int -3 \, dt}\).
Multiply both sides of the differential equation by the integrating factor \(\mu(t)\) to rewrite the left side as the derivative of a product: \(\frac{d}{dt}[\mu(t) y(t)] = \mu(t) q(t)\).
Integrate both sides with respect to \(t\) to find \(\mu(t) y(t) = \int \mu(t) q(t) \, dt + C\), where \(C\) is the constant of integration.
Use the initial condition \(y(1) = 4\) to solve for the constant \(C\), then solve for \(y(t)\) by dividing both sides by \(\mu(t)\).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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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 y' + p(t)y = q(t). They can be solved using an integrating factor, which simplifies the equation into an exact derivative, allowing integration to find the general solution.
추천 영상:
07:39
Classifying Differential Equations

Integrating Factor Method

This method involves multiplying the differential equation by an integrating factor, typically e^(∫p(t)dt), to rewrite the left side as the derivative of a product. This facilitates direct integration to solve for y(t).
추천 영상:
07:33
Euler's Method

Initial Value Problems (IVP)

An IVP specifies the value of the solution at a particular point, such as y(1) = 4. After finding the general solution, the initial condition is used to determine the unique constant, yielding a specific solution.
추천 영상:
05:03
Initial Value Problems
관련 실천
교과서 질문

33–42. Solving initial value problems Solve the following initial value problems.

p'(x) = 2/(x² + x), p(1) = 0

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교과서 질문

45–46. Harvesting problems Consider the harvesting problem in Example 6.

If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?

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교과서 질문

39–42. Special equations A special class of first-order linear equations have the form a(t)y'(t)+a'(t)y(t)=f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form

a(t)y'(t) + a'(t)y(t) = d/dt (a(t)y(t)) = f(t). 

Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems. 


(t² + 1)y′(t) + 2ty = 3t², y(2) = 8

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교과서 질문

17–18. {Use of Tech} Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function.


The population increases from 50 to 60 in the first month and eventually levels off at 150.

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교과서 질문

Stability of Euler's method Consider the initial value problem y′(t) = −ay, y(0) = 1 where a > 0; it has the exact solution y(t) = e⁻ᵃᵗ, which is a decreasing function.


a. Show that Euler's method applied to this problem with time step h can be written u₀ = 1, uₖ₊₁ = (1 − ah)uₖ for k = 0, 1, 2, ...


b. Show by substitution that uₖ = (1 − ah)ᵏ is a solution of the equations in part (a), for k = 0, 1, 2, ...

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교과서 질문

Explain how to solve a separable differential equation of the form

g(t)y'(y) = h(t)

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