교과서 질문
A family of exponential functions
b. Verify that the arc length of the curve y=f(x) on the interval [0, ln 2] is A(2^a-1) - 1/4a²A (2^-a - 1).
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
9장, 문제 9.1.3
Does the function y(t) = 2t satisfy the differential equation y'''(t) + y'(t) = 2?
검증된 단계별 안내1
Identify the given function: \(y(t) = 2t\).
Compute the first derivative \(y'(t)\) by differentiating \(y(t)\) with respect to \(t\): \(y'(t) = \frac{d}{dt}(2t)\).
Compute the third derivative \(y'''(t)\) by differentiating \(y'(t)\) two more times: \(y'''(t) = \frac{d^2}{dt^2} y'(t)\).
Substitute \(y'(t)\) and \(y'''(t)\) into the differential equation \(y'''(t) + y'(t) = 2\) to check if the equation holds true.
Simplify the expression after substitution and verify whether the left-hand side equals the right-hand side for all \(t\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Higher-Order Derivatives
Higher-order derivatives are derivatives of derivatives, such as the second derivative (rate of change of the rate of change) and the third derivative. In this problem, y'''(t) denotes the third derivative of y with respect to t, which must be computed to verify the differential equation.
추천 영상:
02:42
Higher Order Derivatives
Differential Equations
A differential equation relates a function and its derivatives. To check if a function satisfies a differential equation, substitute the function and its derivatives into the equation and verify if the equality holds for all values of the independent variable.
추천 영상:
07:39Classifying Differential Equations
Derivative of a Linear Function
The derivative of a linear function y(t) = mt + b is a constant m, and higher derivatives of a linear function are zero. This property simplifies evaluating y'(t) and y'''(t) for the given function, aiding in checking the differential equation.
추천 영상:
07:17Linearization
관련 실천
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
23–26. Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t≥0 graph the solution, and determine the first month in which the loan balance is zero.
B′(t) = 0.004B − 800, B(0) = 40,000
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교과서 질문
11–16. Initial value problems Solve the following initial value problems.
y'(x) = −y + 2, y(0) = −2
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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
17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = yeᵗ, y(0) = −1
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