Initial Value Problem / Differential Equation

[enkrypt0r]

Thanks for clicking!

So, I've got a problem here that I'm stuck on. I need to find the general solution to

y' = (y³ + 6y² + 9y)/9

I found this to be

ln|y| + (3/(y+3)) - ln|y+3| = x + c

but I would appreciate it if you would check my work. Anywho, once I have the general solution I need to solve it for y(2) = -3. Obviously, plugging -3 in for y creates a problem in the second and third terms above.

Is there a way to rewrite the general solution or solve it in such a way that it won't contain any undefined terms when evaluated like it does now?

Thanks guys.

EDIT: Shoot, sorry for the non-descriptive title. I accidentally clicked post before finishing it. I meant it to say "[Calc II] Initial Value Problem / Differential Equation." My bad.

 

[Mining_Engr]

I got the same general solution, what do you mean by undefined terms?

 

[enkrypt0r]

I mean that when I plug -3 in for y, I get a zero in the denominator of the second term and ln(0) for the third term, both of which are undefined. I was wondering if there is any way to rewrite this or to solve this differently so -3 will not cause this discontinuity.

 

[tiny-tim]

hi enkrypt0r!

y' = (y³ + 6y² + 9y)/9

erm … look at the equation!

if y = -3, what is y' ? ​

 

[enkrypt0r]

hi enkrypt0r!


erm … look at the equation!

if y = -3, what is y' ? ​

Hi, Tim!

I never thought to look at the original equation! This evaluates to 0 = 0. I'm afraid I don't know what this says about the differential equation. Does this mean that it's undefined, as my attempts to evaluate it have shown?

 

[tiny-tim]

oooh … think! …

if y' = 0, then … ? ​

 

[enkrypt0r]

Ummm... I'm feeling stupid here, haha. Sorry. I'm brainstorming but coming up with nothing beyond the obvious. We could integrate both sides and see that y is a constant, but we already know that. I'm sorry, I'm at a loss.

 

[Mark44]

If y = 0 or y = -3, then y' = 0, which says that y is a constant.

In your work, the tacit assumption is that y != 0 and y != -3.

 

[Mining_Engr]

Yeah, y=0 and y=-3 are "equilibria" solutions to the de. This sort of de is known as autonomous, because there are no t's on the right side of the equation. So, no matter what t is, the slope at y=0 and y=-3 is always zero.

So, why can't we plug into solve the initial value problem?

 

[Mining_Engr]

If you look at the direction field, I think you might find that more than one solution curve would satisfy the initial condition. Thus, there is no particular value of c that would net you a unique solution curve. Someone correct me if this is wrong, because I have never seen this case before! That's my analysis though.

 

[Mark44]

If you look at the direction field, I think you might find that more than one solution curve would satisfy the initial condition. Thus, there is no particular value of c that would net you a unique solution curve. Someone correct me if this is wrong, because I have never seen this case before! That's my analysis though.

For the initial value problem, y' = (1/9)(y³ + 6y² + 9y), y(2) = -3, we see that y' = (1/9)y(y + 3)².

If y = -3, then y' = 0, so y(x) $\equiv$ C. Since y(2) = -3, C = -3, so the solution to the IVP is y(x) $\equiv$ -3. Are you saying that you think there are other solutions to this initial value problem?

 

[tiny-tim]

hi enkrypt0r!

(just got up :zzz:)

the solution comes in three parts …

y = -3 (for all t) is a solution

y = 0 (for all t) is a solution

and the other solutions are all the solutions of

dy/(y³ + 6y² + 9y) = dt/9

(you multiplied by 0/0 when you got that equation, if y = -3 or 0, so you illegitmately lost those solutions )

if you draw a graph of some specimen solutions, you'll probably find that they "asymptote up to" the vertical lines at -3 and 0,

and (i'm guessing here ) that one of y = 0 and v = -3 is stable and the other is unstable ​