# How to find unknown variable in the below equation?

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I wish to solve the equation as shown below:

So, I made a new variable "A1" as coded below with i/p parameters.

r = 3.88;

Vau = 43;

theta_u = 71.8;

gamma = 5/3;

Vsu = 47.45;

syms Unu

A1 = double(solve(((Unu^2 - (r * Vau^2 *(cosd(theta_u))^2))^2) * (Unu^2 - ((2 * r * Vsu^2)/(r+1-gamma*(r-1)))) - ((r * (sind(theta_u)^2) * Unu^2 * Vau^2) *((2 * r - gamma * (r-1))*(Unu^2)/(r+1-gamma(r-1)) - r * Vau^2 * (cosd(theta_u))^2))))

A1 shows error like:

Subscript indices must either be real positive integers or logicals.

Error in IPS_P2 (line 333)

A1 = double(solve(((Unu^2 - (r * Vau^2 *(cosd(theta_u))^2))^2) * (Unu^2 - ((2 * r * Vsu^2)/(r+1-gamma*(r-1)))) - ((r * (sind(theta_u)^2) * Unu^2 * Vau^2) *((2 * r - gamma * (r-1))*(Unu^2)/(r+1-gamma(r-1)) - r * Vau^2 * (cosd(theta_u))^2))))

Can anyone please help me, if there is any error in coding equation?

##### 0 Comments

### Accepted Answer

Alan Stevens
on 1 Sep 2020

The equation is a cubic in Unu^2, so the following uses roots to find solutions:

r = 3.88;

Vau = 43;

theta_u = 71.8;

gamma = 5/3;

Vsu = 47.45;

A = r*Vau^2*cos(theta_u)^2;

B = 2*r*Vsu^2/(r+1-gamma*(r-1));

C = r*Vau^2*sin(theta_u)^2;

D = (2*r - gamma*(r-1))/(r+1 - gamma*(r - 1));

% Let x = Unu^2

% (x - A)^2*(x - B) - C*x*(D*x - A) = 0

% (x/A - 1)^2*(x/A - B/A) - C/A*(x/A)*(D*x/A - 1) = 0

% Let y = x/A

% (y^2 - 2y + 1)*(y - B/A) - D*C/A*y^2 + C/A*y = 0

% Expand and collect like terms to get:

% y^3 -(2 + B/A + D*C/A)y^2 + (1 + 2*B/A + C/A)y - B/A = 0

p = [1; -(2 + B/A + D*C/A); (1 + 2*B/A + C/A); -B/A];

y = roots(p);

x = A*y; % reconstruct x

Unu = sqrt(x); % reconstruct Unu

disp(y)

disp(x)

disp(Unu)

% Check (f should be zero)

f = polyval(p,y);

disp(f)

##### 6 Comments

Alan Stevens
on 3 Sep 2020

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