Math Problem Statement

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Solution

To solve this problem, let's break down the given conditions and then prove the statement step by step.

Given:

  1. f(y)f(y) is continuously differentiable on (,+)(-\infty, +\infty).
  2. f(y)<0f(y) < 0 for y0y \neq 0.

We need to prove that for the differential equation dydx=f(y),\frac{dy}{dx} = f(y), the solution y=y(x)y = y(x) with the initial condition y(x0)=y0y(x_0) = y_0 exists on the interval [x0,+)[x_0, +\infty) and that limx+y(x)=0.\lim_{x \to +\infty} y(x) = 0.

Proof:

  1. Existence and Uniqueness: Since f(y)f(y) is continuously differentiable, it satisfies the conditions of the Picard-Lindelöf theorem. Thus, for the initial value problem dydx=f(y)\frac{dy}{dx} = f(y) with y(x0)=y0y(x_0) = y_0, there exists a unique solution y=y(x)y = y(x) in some interval around x0x_0.

  2. Behavior of f(y)f(y): Given that f(y)<0f(y) < 0 for y0y \neq 0, yy decreases when y0y \neq 0. Therefore, if y(x)y(x) starts at y00y_0 \neq 0, it will move towards 0 as xx increases.

  3. Asymptotic Behavior: To show that limx+y(x)=0\lim_{x \to +\infty} y(x) = 0, consider the behavior of the solution as x+x \to +\infty.

    Assume yy does not approach 0. Then there exists some ϵ>0\epsilon > 0 and a sequence xn+x_n \to +\infty such that y(xn)ϵ|y(x_n)| \geq \epsilon. However, since f(y)<0f(y) < 0 for y0y \neq 0, yy should be decreasing and bounded away from 0, leading to a contradiction because y(x)y(x) must cross any positive threshold infinitely often. This contradiction implies that y(x)y(x) must approach 0 as x+x \to +\infty.

  4. Detailed Analysis: By integrating the differential equation, we get: xx0=y0ydyf(y).x - x_0 = \int_{y_0}^{y} \frac{dy}{f(y)}.

    Since f(y)<0f(y) < 0 for y0y \neq 0, the integral y0ydyf(y)\int_{y_0}^{y} \frac{dy}{f(y)} is well-defined and finite as yy approaches 0. Hence, x+x \to +\infty as y0y \to 0, implying that y(x)y(x) approaches 0 as x+x \to +\infty.

Therefore, the solution y=y(x)y = y(x) with the initial condition y(x0)=y0y(x_0) = y_0 exists on the interval [x0,+)[x_0, +\infty) and satisfies limx+y(x)=0\lim_{x \to +\infty} y(x) = 0.

Would you like any more details or have any questions about this proof? Here are some questions you might consider:

  1. Can you explain the Picard-Lindelöf theorem in more detail?
  2. How does the condition f(y)<0f(y) < 0 influence the behavior of the solution?
  3. Can you show the integration steps in more detail?
  4. What are other methods to prove the asymptotic behavior of differential equations?
  5. How would the solution change if f(y)>0f(y) > 0 for y0y \neq 0?

Tip: Always ensure to verify the conditions for the existence and uniqueness theorems before applying them to differential equations.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Existence and Uniqueness Theorem
Asymptotic Behavior

Formulas

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Theorems

Picard-Lindelöf theorem

Suitable Grade Level

Advanced Undergraduate