교과서 질문
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
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Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
9장, 문제 9.2.32
Solve the Bernoulli equations in Exercises 29–32.
x²y' + 2xy = y³
검증된 단계별 안내1
Rewrite the given differential equation \(x^{2}y' + 2xy = y^{3}\) in the standard Bernoulli form. First, isolate \(y'\) by dividing the entire equation by \(x^{2}\):
\[y' + \frac{2}{x}y = \frac{y^{3}}{x^{2}}\]
Identify the Bernoulli equation parameters: it has the form
\[y' + P(x)y = Q(x)y^{n}\]
where \(P(x) = \frac{2}{x}\), \(Q(x) = \frac{1}{x^{2}}\), and \(n = 3\).
Make the substitution \(v = y^{1-n} = y^{1-3} = y^{-2}\). Then compute \(v'\) in terms of \(y\) and \(y'\):
\[v = y^{-2} \implies v' = -2y^{-3}y'\]
Express \(y'\) from the substitution:
\[y' = -\frac{1}{2} y^{3} v'\]
Substitute \(y'\) and \(y^{3}\) back into the original equation rewritten in step 1 to get a linear differential equation in terms of \(v\).
After substitution, simplify the resulting equation to the form
\[v' + p(x)v = q(x)\]
where \(p(x)\) and \(q(x)\) are functions of \(x\). Then solve this linear first-order ODE using an integrating factor.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Bernoulli Differential Equation
A Bernoulli differential equation has the form y' + P(x)y = Q(x)y^n, where n is any real number other than 0 or 1. It is nonlinear due to the y^n term but can be transformed into a linear equation using an appropriate substitution.
추천 영상:
07:39
Classifying Differential Equations
Substitution Method for Bernoulli Equations
To solve a Bernoulli equation, use the substitution v = y^(1-n), which converts the nonlinear equation into a linear differential equation in terms of v. This allows the use of standard methods for linear equations to find the solution.
추천 영상:
07:33Euler's Method
Solving Linear First-Order Differential Equations
Once transformed, the equation becomes linear and can be solved using an integrating factor, μ(x) = exp(∫P(x) dx). Multiplying through by μ(x) simplifies the equation to an exact derivative, enabling integration and solution for v, and subsequently y.
추천 영상:
06:06Solving Separable Differential Equations
관련 실천
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' = 2xy + 2y, y(0) = 3, dx = 0.2
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교과서 질문
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
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교과서 질문
Show that the solution of the initial value problem
y' = x + y, y(x₀) = y₀
is
y = -1 -x + (1 + x₀ + y₀) exp(x-x₀).
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교과서 질문
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
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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