How do I calculate the slope of an equation if I don't have any initial data?
du/dt + (1/2 * (1 + u)^2) = 0
The slope is therefore
dx/dt = a(u) = f'(u) = 1 + u
With no initial data how do I calculate u(x,t)?
Calculating anything without initial data is veeeeeery troublesome =)
du/dt + 1/2*(1+u)^2 = 0
du/dt+ 1/2*(1^2+2u+u^2) = 0
du/dt +1/2 + u + u^2 = 0
du/dt + u = -u^2 - 1/2 this gives you ke^-x
Set
$\displaystyle u = ax^2 + bx + c $ gives you uŽ= 2ax +b
$\displaystyle 2ax +b +ax^2 +bx +c = -u^2 -1/2$ gives you a = -1
$\displaystyle
2ax +bx = 2*(-1) +b = 0
$ b = 2
$\displaystyle b + c = -1/2$ gives you c = -2,5
$\displaystyle U = Ke^(-x) -x^2 +2x -2.5$
To determine K, I think you need a value somewhere,
If you plot that graph for different values of K you will see that the slopes have different angles exept for the x value for maximum/minimum point, is that the value you are trying to find? Otherwise you need some initial data to determine K. ( I think)
hope it was somewhat helpful.