Math Is Fun Forum

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.

Hi all just one hopefully quick question which is really bothering me

Solve sin2x=sinx over [0,2pi]

sin2x=2sinxcosx from double angle formula

2sinxcosx=sinx
divide both sides by 2sinx
cosx=sinx/2sinx
cosx=1/2
x= pi/3 or 5pi/3

this is wrong however, what is it that is incorrect?

Hi all, I'm getting frustrated with dealing with applications of differential equations.
With regards to when to add in the c constant, and how you are to go about manipulating it.
Take for example this applications question;

A city with population P, at time t years after a certain date, has a population which increases at a rate proportional to the population at that time.
a) i) Set up a DE to describe this situation;
Let P= Population
    t= time (years)
dP/dt proportional to p
hence Dp/dt=kP where k>0.

ii) Solve to obtain a general solution.
dP/dt=kP
dt/dP=1/kP
t=∫1/kP dP
this is where the problem arises;I want to solve the integral for P which will help with later questions.
My teacher goes about it this way.

t= 1/k integral(1/P) dP
tk= loge|P|+c      oh and if you solve for t here would you get; t=1/kloge|P|+c/k?
tk-c=loge|P|
e^(tk-c)=P

Alternatively I tried this approach;
t=1/k∫(1/P) dP
t=1/kloge|P|+c
t-c=1/kloge|P|
k(t-c)=loge|P|
e^k(t-c)=P
which is correct? and why?

I have solved for P because the next question asks to find the population after x years given other data.