Help simplifying a solution to a differential equation

[Raziel2701]

So I'm in the process of solving the following differential equation:

$$\frac{dy}{dx}=2y^2 +xy^2$$

with initial condition y(0)=1

I worked it out until I got to the following equation:

$$y^2 +y =x^2 -4$$

So now my problem is that I can't isolate y as a function of x in order to move on with the problem and determine where the solution attains its minimum value.

I've tried completing the square and I noticed that the RHS(right hand side) is a difference of squares and I expanded that out but I have no clue how to isolate that y. Now I'm tired and desolate, and I'd like to be pointed back on the right direction.

It's frustrating to be thwarted by algebraic manipulations. :(

 

[bennyska]

so i could be totally wrong on this, I'm here myself to get help with simple differential equations, but here goes:

so you have

dy/dx = 2y² + x*y², with y(0) = 1

try factoring it as dy/dx = y² ( 2 + x)
then you separate the variables.
does that help?

p.s. could you maybe show how you got "the following equation"? i have no idea where you got that. i think you may have made a mistake, but like i said, I'm here for help myself.

 

[HallsofIvy]

I agree with bennyska. I get a completely different solution that you do. Please show us how you got that.

 

[Raziel2701]

Man I got mixed up. The original differential equation is:
$$\frac{dy}{dx}=\frac{2x}{1+2y}$$ with initial condition y(2)= 0

I've been working on this mountain load of homework last night that I got my equations all wrong, I'm sorry.

So with that in mind, after I separate and integrate I get:

$$y^2 +y=x^2-4$$

Now I'd like to know how to isolate y, because I haven't been able to, I am not sure what I have to expand or what. I'm just, nonplussed.

 

[MarcMTL]

Man I got mixed up. The original differential equation is:
$$\frac{dy}{dx}=\frac{2x}{1+2y}$$ with initial condition y(2)= 0

I've been working on this mountain load of homework last night that I got my equations all wrong, I'm sorry.

So with that in mind, after I separate and integrate I get:

$$y^2 +y=x^2-4$$

Now I'd like to know how to isolate y, because I haven't been able to, I am not sure what I have to expand or what. I'm just, nonplussed.

Again, not sure what you did to get that answer. Where did the -4 come from on the dx side?

You should separate $$2xdx = (1+2y)dy$$

 

[Raziel2701]

Yes I separated like you did and integrated, obtaining:

$$y+y^2=x^2 +c$$

The initial condition is y(2) = 0, the one I posted in my first post is from a different problem I got mixed in. Thus plugging 2 for x, and 0 for all y, I get that c=-4, which I then substitute into the equation we have so far and I get:

$$y+y^2=x^2 -4$$

Which I've tried to manipulate to isolate y, but have been unsuccessful so far.

 

[Mark44]

Write your equation as y² + y - x² + 4 = 0.

This is quadratic in y, with a = 1, b = 1, and c = 4 - x². Just plug these into the quadratic formula to get two solutions for y(x).

 

[Raziel2701]

I was never taught I could do this kind of stuff with the quadratic equation. I feel ripped off :p

Thanks Mark44!