Math Is Fun Forum

glenn101

Member

Registered: 2008-04-02

Posts: 108

Seperable Differential Equations Conceptual Help

Hey all,
I'm having difficulty understanding several concepts with regards to solutions of seperable differential equations.

The S.D.E I am talking about is of the form f'(x)=f(y)*f(x).

From my understanding, a solution of an initial value problem must satisfy BOTH the D.E and the initial conditions given in the stated problem, and this is where my confusion begins.

Apparently, a constant solution satisfies f(y)=0 and if the initial conditions y-value matches the constant solution found, that is your solution to your S.D.E initial value problem.

Ok so for an example f'(x)=3*x*y-6*x with y=2 when x=1.
f'(x) = 3*x(y-2)
so then f(y) = y-2
which means that f(y)=0 for y=2.

now, I understand that when x=1, y=2 will be satisfied by y=2 but what I don't get is how on earth y=2 satisfies the D.E? as f'(x)=0 not 3*x(y-2).
How can it be called a constant 'solution' if it only satisfied the initial conditions?

This follows onto my next question, if I changed the initial conditions of the previous question to say y=1 when x=1
why can we then say that the solution must lie beneath the line y=2? I've heard that solutions can't cross, but if by solutions they mean solutions to IVP's well then, how can y=2 be a constant solution in this case? it doesn't satisfy the initial conditions OR the D.E.

I know also, that dividing by zero comes into play when trying to seperate the D.E. because if y satisfies the initial condition while also the constant solution, you would be dividing by zero. But I'm still quite unclear with all this, any help would be greatly appreciated.

Thanks,
Glenn101.

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"If your going through hell, keep going."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Seperable Differential Equations Conceptual Help

Your example is a DE of the form

where

and

. The general solution is

As you can see the initial conditions are impossible as you can’t have

on the LHS.

Last edited by JaneFairfax (2010-08-16 00:27:17)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Seperable Differential Equations Conceptual Help

Actually, if y = 2 then f is just the constant function f(x) = 2 and there is nothing to solve.