matlab, how do i write a statement that will give me time on xaxis from y=0.3

Question

x=[0:.01:10];
y=(x.^2).*(exp(-x));

plot(x,y), grid
y1=max(y);

xlabel(\'TIME\');
ylabel(\'CONCENTRATION IN BLOOD\');
title(\'CONCENTRATIN OF SUSTANCE IN BLOOD vs TI         


        

Answer 1

If you have the symbolic toolbox in MATLAB, you can do the following

syms x
x=solve('x^2*exp(-x)=y')

x=
  (-2)*lambertw(k, -((-1)^l*y^(1/2))/2)

Here lambertw is the solution to y=x*exp(x), which is available as a function in MATLAB. So you can now define a function as,

t=@(y,k,l)(-2)*lambertw(k, -((-1)^l*y^(1/2))/2)

lambertw is a multivalued function with several branches. The variable k lets you choose a branch of the solution. You need the principal branch, hence k=0. l (lower case L) is just to pick the appropriate square root of y. We need the positive square root, hence l=0. Therefore, you can get the value of t or the time for any value of y by using the function.

So using your example, t(0.3,0,0) gives 0.8291.

EDIT

I forgot that there are two branches of the solution that give you real outputs (gnovice's answer reminded me of that). So, for both solutions, use

t(0.3,[0,-1],0)

which gives 0.8921 and 3.9528.