How can I solve the following system in Maple for $S_1$ and $S_{i+1}$? I have the code written, but it is giving me nothing as output.

eq1 := Q-A*S[1]*C/X+B*D*(sum(S[j], j = 2 .. i+1))/Y-r[1]*S[1] = 0;
eq2 := A*S[i-1]*C/X-A*S[i]*C/X-B*S[i]*D/Y-r[1]*S[i] = 0;
eq3 := A*S[i]*C/X-B*S[i+1]*D/(Y+S[i+1])-r[2]*S[i+1] = 0;
solve({eq1, eq2, eq3}, {S[1], S[i+1]});

The only non-constants in the system are the $S_j$'s for $j = 1, \ldots, i+1$.

Here is the system in math mode:
$$Q-\frac{AC}{X}S_1-r_1 S_1 +\frac{BD}{Y}\sum\limits_{j=2}^{i+1}S_j = 0 \\
\frac{AC}{X}S_{i-1} - \frac{AC}{X}S_i - \frac{BD}{Y}S_i - r_1 S_i = 0\\
\frac{AC}{X}S_i - \frac{BD S_{i+1}}{Y+S_{i+1}} - r_2 S_{i+1} = 0 $$

I have a head start on it by hand, but it's too cumbersome to complete.

Basically, my strategy is to solve for $S_{i+1}$ as a function of $S_1$ and some constants. Then to plug in $S_2, \ldots, S_{i+1}$ in the summation and solve for $S_1$ as a function of constants. Then I would obtain $S_{i+1}$ as a function of constants itself.

Thanks for any help.

I need to show that the following expression,
a^3b-a^3c+a^3z+a^3x+a^3y-a^2bx+a^2by+a^2cx-a^2cy-a^2zx+a^2zy-a^2x^2+a^2y^2-abcz-abcx-aczx-acx^2+b^2c^2+2bc^2x+c^2x^2-b^2c-2bcx-cx^2,

is positive

given that:

1. a,b,c,x,y,z are positive real numbers

2. a>b+x

3. c<b+y

I know a priori that the expression is indeed positive, but I do not know how to show it, or how to use Maple to do it?

Specifically, how can I use Maple to **partially factorize** the expression in terms of the expressions a-b-x and c-b-y?

Thanks for any help.

I have generated an 8x8 Jacobian, containing a few variables and several zeros as elements. I would like to translate this to LaTeX code. How can I first simplify what I have, to make it tractable?

Here is the Maple code:

eq_1 := Lambda-mu*S-(beta*(H+C+C1+C2)*S+tau*(T+C)*S)/(S+T+H+C+C1+C2+C1M+C2M);
eq_2 := (-beta*(H+C+C1+C2)*T+tau*(T+C)*S)/(S+T+H+C+C1+C2+C1M+C2M)-(mu+mu[T])*T;
eq_3 := (beta*(H+C+C1+C2)*S-tau*(T+C)*H)/(S+T+H+C+C1+C2+C1M+C2M)-(mu+mu[A])*H;
eq_4 := (beta*(H+C+C1+C2)*T+tau*(T+C)*H)/(S+T+H+C+C1+C2+C1M+C2M)-(mu+mu[T]+mu[A]+lambda[T])*C;
eq_5 := lambda[T]*C-(mu+mu[A]+rho[1]+eta[1])*C1;
eq_6 := rho[1]*C1-(mu+mu[A]+rho[2]+eta[2])*C2;
eq_7 := eta[1]*C1-(mu+rho[1]+gamma)*C1M;
eq_8 := eta[2]*C2+rho[1]*C1M-(mu+rho[2]+gamma*rho[1]/(rho[1]+rho[2]))*C2M;
J := VectorCalculus:-Jacobian([eq_1, eq_2, eq_3, eq_4, eq_5, eq_6, eq_7, eq_8], [S, T, H, C, C1, C2, C1M, C2M]);

JQDFE := eval(J, [S = Lambda/(beta-mu[A]), T = 0, H = Lambda*(beta/(mu+mu[A])-1)/(beta-mu[A]), C = 0, C1 = 0, C2 = 0, C1M = 0, C2M = 0]);

Thanks.

Hello,

      I would like to solve a system of 9 nonlinear equations, with the constraints on all 9 variables to be that they are nonnegative. How can I do this?

My code is below - I am trying NLPSolve and have tried solve, but am getting stuck.

with(Optimization);

restart; eq1 := 531062-S/(70*365)-(.187*(1/365))*(H+C+C1+C2)*S/N = 0;eq2 := (4/365*(T+C))*S/N-(.187*(1/365))*(H+C+C1+C2)*T/N-(1/(70*365)+1/(5*365))*T = 0; eq3 := (.187*(1/365))*(H+C+C1+C2)*S/N-(4/365)(T+C)*H/N-(1/(70*365)+1/(4*365))*H = 0; eq4 := (.187*(1/365))*(H+C+C1+C2)*T/N+(4/365*(T+C))*H/N-(1/(70*365)+3/(8*365)+.2*(1/365)+.1)*C = 0; eq5 := .1*C-(1/(70*365)+1/(4*365)+1/60+.5)*C1 = 0; eq6 := (1/60)*C1-(1/(70*365)+1/(4*365)+1/210+.5)*C2 = 0; eq7 := .5*C1-(1/(70*365)+1/60+0.1e-2)*CT1 = 0; eq8 := .5*C2-(1/(70*365)+1/210+(1/9)*(0.1e-2*7))*CT2+(1/60)*CT1 = 0; eq9 := N-S-T-H-C-C1-C2-CT1-CT2 = 0; soln := NLPSolve({eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9}, {C, C1, C2, CT1, CT2, H, N, S, T}, assume = nonnegative);

I am numerically solving a nonlinear system of nine equations. How long can I expect it to take?

I have run it for 30 minutes and it has not solved yet.

Here is the system of equations:

0=Lambda-mu.*S-beta.*(H+C+C1+C2).*(S./N)-tau.*(T+C).*(S./N);

0=tau.*(T+C).*(S./N)-beta.*(H+C+C1+C2).*(T./N)-(mu+mu_T).*T;

0=beta.*(H+C+C1+C2).*(S./N)-tau.*(T+C).*(H./N)-(mu+mu_A).*H;

0=beta.*(H+C+C1+C2).*(T./N)+tau.*(T+C).*(H./N)-(mu+psi.*mu_A+mu_T+lambda_T).*C;

0=lambda_T.*C-(mu+mu_A+rho_1+eta_1).*C1;

0=rho_1.*C1-(mu+mu_A+rho_2+eta_2).*C2;

0=eta_1.*C1-(mu+rho_1+gamma).*CT1;

0=eta_2.*C2-(mu+rho_2+gamma.*(rho_1)./(rho_1+rho_2)).*CT2+(rho_1).*CT1;

0 = N - S - T - H - C - C1 - C2 - CT1 - CT2;

and I have numeric values for Lambda, beta, tau, mu, mu_T, mu_A, rho_1, rho_2, psi, gamma. The only parameters left are eta_1, eta_2.

Thank you.