First Order Differential Equation

[tomeatworld]

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:

$$\frac{y'-1}{2y+1}$$ = $$\frac{1}{x}$$ 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:

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

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

 

[Mark44]

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:

$$\frac{y'-1}{2y+1}$$ = $$\frac{1}{x}$$ 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
$$e^{\int \frac{-2}{x}dx}$$

 

[tomeatworld]

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(($$Ccos(\sqrt{2}t) + Dsin(\sqrt{2}t)$$))
ii) I don't really understand this question, but from what I can gather after brief research:
Basis{(e^-t$$cos\sqrt{2}t$$),(e^-t$$sin(\sqrt{2}t)$$))}. 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?

 

[Mark44]

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(($$Ccos(\sqrt{2}t) + Dsin(\sqrt{2}t)$$))
ii) I don't really understand this question, but from what I can gather after brief research:
Basis{(e^-t$$cos\sqrt{2}t$$),(e^-t$$sin(\sqrt{2}t)$$))}. 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.