Differential equation dy/dx = 2y-4x

[nns91]

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 ?

 

[rock.freak667]

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.

 

[nns91]

a. Got cha.

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

 

[rock.freak667]

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?

 

[nns91]

Yeah. so is the integrating factor only e^2 ?
I got the final equation as y=-2x^2. Am I right ?

 

[nns91]

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 ?