Differential equation dy/dx = 2y-4x

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Homework Help Overview

The discussion revolves around a differential equation of the form dy/dx = 2y - 4x. Participants are exploring potential solutions and the implications of specific conditions related to the equation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to find a specific value of b that makes y=2x+b a solution to the differential equation. There are questions about the correctness of their calculations and the simplicity of the problem.
  • In part b, there are discussions about finding the function g that satisfies the differential equation with an initial condition and whether it has a local extremum at (0,0). Participants are questioning the nature of critical points and how to proceed with integration.
  • Some participants suggest using an integrating factor to solve the differential equation, while others express uncertainty about their results.

Discussion Status

The discussion is ongoing, with various participants sharing their thoughts and attempts. Some guidance has been offered regarding the use of derivatives and integrating factors, but there is no clear consensus on the solutions or methods being discussed. Multiple interpretations of the problem are being explored.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the information they can share or the methods they can use. There is also a mention of testing endpoints in a series convergence problem, indicating a broader mathematical context.

nns91
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Homework Statement



Consider the differential equation dy/dx = 2y-4x

a. Find the value of b for which y=2x+b is a solution to the given differential equation

b.Let g be the function that satisfies the given differential equation with the initial condition g(0)=0. Does the graph of g have a local extremum at the point (0,0) ? If so, is the point a local maximum or minimum

Homework Equations



integration, derivative

The Attempt at a Solution



a. So I choose point (1,0) and plug in y=2x+b and solve for b=1. Then I have y=2x+1. AM I right ? It seems too easy to be right. Is it harder than I thought ?

b. So do I have to integrate ? I found that (0,0) is a critical point. How do I move on ?
 
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For part a) what you need to do is find dy/dx from your given solution and then substitute y in the differential equation and find b. Can you do that?

It would be beneficial to solve the differential equation.
 
a. Got cha.

b. is g(x)= 2x+1 ? Am I right ? It sounds wrong
 
nns91 said:
a. Got cha.

b. is g(x)= 2x+1 ? Am I right ? It sounds wrong

Do you know how to solve 1st order DEs of the form
\frac{dy}{dx}+P(x) y=Q(x)

via an integrating factor?
 
Yeah. so is the integrating factor only e^2 ?
I got the final equation as y=-2x^2. Am I right ?
 
Also a question about series:

Given a McLauren series: (2x)^n+1 / (n+1)

(a). Find interval of convergence.

So I used ratio test and found that -1/2 <x<1/2. I am testing the end point. At x=1/2, the series will be 1/(n+1) and at x=-1/2, series is (-1)^n+1 / (n+1). How do I prove whether or not they are divergent or convergent. Does 1/ (n+1) converge to 0 ?
 

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