First Order Differential Equation

  • #1

Homework Statement


Find the general solution for: x y' - 2y = x +1 (x>0)

Homework Equations


None

The Attempt at a Solution


I have literally no idea how to start this. I've tried seperating variables but ended up with:

[tex]\frac{y'-1}{2y+1}[/tex] = [tex]\frac{1}{x}[/tex] but that isn't solvable due to the y'-1 (at least I don't know how if it is). Any help is greatly appreciated!

Edit: Ok, so I've used integrating factors. If you take the integrating factor to be x-2 (i can show how if needed) and then use that, I find:

[tex]\frac{d}{dx}(x^{-2}y) = x^{-2}+ x^{-3}[/tex] so the equation integrates to:

[tex] y = cx^{2}-x-\frac{1}{2}[/tex] does that seem about right? We have disagreement in our group :/
 
Last edited:

Answers and Replies

  • #2
35,129
6,876

Homework Statement


Find the general solution for: x y' - 2y = x +1 (x>0)

Homework Equations


None

The Attempt at a Solution


I have literally no idea how to start this. I've tried seperating variables but ended up with:

[tex]\frac{y'-1}{2y+1}[/tex] = [tex]\frac{1}{x}[/tex] but that isn't solvable due to the y'-1 (at least I don't know how if it is). Any help is greatly appreciated!
An integrating factor will work.

Rewrite the DE as y' - (2/x)y = 1 + 1/x

The integrating factor is
[tex]e^{\int \frac{-2}{x}dx}[/tex]
 
  • #3
Thanks! Just edited above the work I managed with that! Does that seem right?

Ok, rather than start a new thread (which I can do if it would be easier), I thought I'd put in the new question I have here:

Homework Statement


i) Find the general solution for y'' + 2y' + 3y =0
ii) Give the basis for the vector space of all solutions of this equation.
iii) Describe the behaviour of all solutions as t tends to infinity
iv) If this equation models a mass-spring system with no external force, state whether the motion is underdamped, critically damped or overdamped.

Homework Equations


None

The Attempt at a Solution


i) Easy enough:
y(t)=e-t(([tex]Ccos(\sqrt{2}t) + Dsin(\sqrt{2}t)[/tex]))
ii) I don't really understand this question, but from what I can gather after brief research:
Basis{(e-t[tex]cos\sqrt{2}t[/tex]),(e-t[tex]sin(\sqrt{2}t)[/tex]))}. Can someone explain this question please?
iii) as t tends to infinity, y(t) tends to 0, due to the exponential decay.
iv) I'm not really sure here. All my notes on mass-spring systems don't really cover differential solutions when I haven't worked from a frequency point-of-view. I think it shows an underdamped system.

Are those about right?
 
Last edited:
  • #4
35,129
6,876
Thanks! Just edited above the work I managed with that! Does that seem right?

Ok, rather than start a new thread (which I can do if it would be easier), I thought I'd put in the new question I have here:

Homework Statement


i) Find the general solution for y'' + 2y' + 3y =0
ii) Give the basis for the vector space of all solutions of this equation.
iii) Describe the behaviour of all solutions as t tends to infinity
iv) If this equation models a mass-spring system with no external force, state whether the motion is underdamped, critically damped or overdamped.

Homework Equations


None

The Attempt at a Solution


i) Easy enough:
y(t)=e-t(([tex]Ccos(\sqrt{2}t) + Dsin(\sqrt{2}t)[/tex]))
ii) I don't really understand this question, but from what I can gather after brief research:
Basis{(e-t[tex]cos\sqrt{2}t[/tex]),(e-t[tex]sin(\sqrt{2}t)[/tex]))}. Can someone explain this question please?
Yes, that's the basis. The solution space is a vector (or function) space that is spanned by those two functions. Every solution of the given DE is a linear combination of those two functions.
iii) as t tends to infinity, y(t) tends to 0, due to the exponential decay.
iv) I'm not really sure here. All my notes on mass-spring systems don't really cover differential solutions when I haven't worked from a frequency point-of-view. I think it shows an underdamped system.

Are those about right?
Yes. For iv, over-, under-, and critically damped relate directly to the discriminant in the solution of the characteristic equation (which in this case is r2 + 2r + 3 = 0).

The discriminant D is b4 - 4ac. If D < 0, the system is underdamped, since there is still a fair amount of oscillation. If D > 0, there is no oscillation, and the system is overdamped. If D = 0, the system is critically damped.
 

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