Need to solve equations which take vector inputs. I have tried with matlab solve() but I'm getting zeros for both tau_new & b

Question

rc=1;

x=linspace(0,1,1000);

R=exp(1).*x.*rc;

b0=.3614/sqrt(2);

eqn1=(b./(8*pi)).(1./R).(log(R./rc)+1)+(b./(2*b0))-tau_new==0,

eqn2=(b./(2*pi)).(1./R).(log(R./rc))+(b./(b0))-tau_new==0];

vars=[tau_new b];

[solv, solu] = solve(eqns, vars)

The goal is to have an expression relating tau_new with R i.e. tau_new=f(R) also b=f(R). Can someone help please?

Answer 1

You have

$$ \left\{ \begin{array}{lcr} \frac{\left(\log \left(\frac{R}{\text{rc}}\right)+1\right) b}{8 \pi R}+\frac{b}{2 \text{b0}}-\tau &=&0 \\ \frac{\log \left(\frac{R}{\text{rc}}\right) b}{2 \pi R}+\frac{b}{\text{b0}}-\tau &=&0 \\ \end{array} \right. $$

or

$$ \left\{ \begin{array}{lcr} G_1(R) b-\tau &=&0 \\ G_2(R)b-\tau &=&0 \\ \end{array} \right. $$

or

$$ \left( \begin{array}{cc} G_1(R) & -1 \\ G_2(R) & -1 \end{array} \right) \left( \begin{array}{c} b \\ \tau \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right) $$

with $G_1(R) \ne G_2(R)$

so $\det \left( \begin{array}{cc} G_1(R) & -1 \\ G_2(R) & -1 \end{array} \right) = G_2(R)-G_1(R) \ne 0$ and the only solution for this homogeneous system is $b = \tau = 0$