교과서 질문
Integral Equations
In Exercises 7–12, write an equivalent first-order differential equation
and initial condition for y.
y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt
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Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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.AAE.7
Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.
(x.exp(y/x) + y)dx - x dy = 0
검증된 단계별 안내1
Identify the given differential equation: \((x \cdot \exp(y/x) + y) \, dx - x \, dy = 0\).
Rewrite the equation in the form \(M(x,y) \, dx + N(x,y) \, dy = 0\), where \(M(x,y) = x \exp(y/x) + y\) and \(N(x,y) = -x\).
Check if the equation is homogeneous by verifying if \(M(tx, ty)\) and \(N(tx, ty)\) are homogeneous functions of the same degree. Here, substitute \(x\) by \(tx\) and \(y\) by \(ty\) in \(M\) and \(N\) and simplify.
If the equation is homogeneous, introduce the substitution \(v = \frac{y}{x}\), which implies \(y = vx\) and \(dy = v \, dx + x \, dv\).
Rewrite the original differential equation in terms of \(v\) and \(x\) using the substitution, then separate variables or rearrange to solve the resulting equation.
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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.
Homogeneous Differential Equations
A differential equation is homogeneous if it can be expressed so that each term is a function of the ratio y/x (or x/y). This allows the equation to be simplified by substituting v = y/x, reducing it to a separable form. Recognizing and rewriting the equation in this form is essential for solving it.
추천 영상:
07:39
Classifying Differential Equations
Substitution Method (v = y/x)
The substitution v = y/x transforms the original variables into a single variable, simplifying the differential equation. This substitution converts the equation into one involving v and x, often making it separable and easier to integrate. Understanding how to apply and manipulate this substitution is key to solving homogeneous equations.
추천 영상:
06:30Disk Method Using y-Axis
Separable Differential Equations
A separable differential equation can be written as a product of a function of x and a function of y, allowing variables to be separated on opposite sides of the equation. After substitution, the homogeneous equation often becomes separable, enabling integration with respect to each variable independently to find the solution.
추천 영상:
06:06Solving Separable Differential Equations
관련 실천
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Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.
y' = y/x + cos ((y-x)/x)
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교과서 질문
Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.
(x²+y²)dx + xy dy = 0
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교과서 질문
Show that (0, 0) and (c/d, a/b) are equilibrium points. Explain the meaning of each of these points.
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교과서 질문
Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.
(x sin y/x - y cos y/x)dx + (x cos y/x) dy = 0
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교과서 질문
In Exercises 1–22, solve the differential equation.
x dy - (x⁴ - y) dx = 0
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