Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equations

I have the following 4 governing equations

i) V1(t) = a.Id(t)^2 + Rd.Id(t)

ii) V1(t) = V(t) - R.I(t)

iii) Ic(t)= C.dV(t)/dt

iv) I(t) = Ic(t) + Id(t)

The values of a, Rd, R and C are constant.

And I need to find an equation in terms of V on the LHS and I on the RHS, but no matter how I look at it I can't find a solution. I've been two days looking at this and I must be missing something. everytime I try to solve it I start to get terms which are the product of I or dI/dt and V or dV/dt

Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

Quote:

Originally Posted by

**munkifisht** V1(t) = a.Id(t)^2 + Rd.Id(t)

Kind of "confusing"...does a.Id(t)^2 mean a times Id (Id = one variable) times t^2 ?

Can we let u = V1, v = Id and w = Rd, and rewrite that equation this way: ut = avt^2 + vwt ?

Then it can be divided through by t, to become: u = avt + vw

????????????

Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

Id(t)^2 simple means that Id which is a function of t is squared.

might be clearer as [Id(t)]^2

Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

I'm just going to rewrite this in a way I think is a little clearer, dump the constant terms (really not that important to this solution) and use the following terms

X, Y, Z, J and K

Where everything is a function of t

I want to combine these equations into a single equation where the X, Y, and Z terms are no longer in the equation and there are only terms of J on the RHS and K on the LHS or visa versa

i) X = Y^2 + Y

ii) Z = dX/dt

iii) J = Z + Y

iv) K = X + J

This should be possible but not matter which way I combine these I get something along the lines of

K = J^2 + ( K'(t) + J'(t) )^2 - 2*J ( K'(t) + J'(t) ) - ( K'(t)+J'(t) )

and obviously this is going to leave you with products of J and Ks that can't be seperated (I need all the K'(t) terms to be over with it's buddy on the LHS). No matter what stratagy I use to avoid this I can't help but get these terms combining.

Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

Quote:

Originally Posted by

**munkifisht** i) X = Y^2 + Y

ii) Z = dX/dt

iii) J = Z + Y

iv) K = X + J

This should be possible but not matter which way I combine these I get something along the lines of

K = J^2 + ( K'(t) + J'(t) )^2 - 2*J ( K'(t) + J'(t) ) - ( K'(t)+J'(t) )

and obviously this is going to leave you with products of J and Ks that can't be seperated (I need all the K'(t) terms to be over with it's buddy on the LHS). No matter what stratagy I use to avoid this I can't help but get these terms combining.

Quite a challenging thingy!

Tried : k = K'(t) , j = J'(t)

to get:

K^2 - 4k^2 + 8k(J - j) = 4J^2 + 4j^2 - 8Jj - 4J - 1

Close, but no ceegar...

For curiosity, tried K = K'(t) , J = J'(t) ; so:

K = J^2 + (K + J)^2 - 2J(K + J) - (K + J) ; leads to:

J = K(K - 2)

And that has endless integer solutions:

J....K

24,-4

15,-3

8,-2

3,-1

0,0

-1,1

0,2

3,3

8,4

Hope that helps....probably not!

Lots of others at this site more advanced than me...maybe one will step in

Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

Quote:

Originally Posted by

**Wilmer** K^2 - 4k^2 + 8k(J - j) = 4J^2 + 4j^2 - 8Jj - 4J - 1

Made a goof; that should be:

K + k(1 - k) + 2k(J - j) = J(J - 2j) + j(j - 1)

Sorry...