# Solve Equation for w(t)

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Mario Braumüller on 30 Jan 2021
Commented: Rik on 28 Feb 2021
This question was flagged by Walter Roberson
I
Rik on 28 Feb 2021
Hello, i tried it with the following code:
syms rho_w w(t) p_u d_0 D_0 l h_0 t kappa m_St m_w rho_L c_w A_Q v(t) h_d lambda zeta n
eqn = (rho_w / 2) * w(t)^2 + p_u == (rho_w / 2) * (d_0 / D_0)^4 * w(t)^2 + ((l - h_0) / (l - h_0 + (d_0 / D_0)^2 * w(t) * t))^kappa + (rho_w / (m_St + m_w - rho_w * (pi / 4) * d_0^2 * w(t) * t)) * (rho_w * (pi / 4) * d_0^2 * w(t)^2 - (rho_L / 2) * c_w * A_Q * v(t)^2) * (h_0 + h_d - (d_0 / D_0)^2 * w(t) * t) + (rho_w / 2) * (d_0 / D_0)^4 * w(t)^2 * (lambda * ((h_0 - (d_0 / D_0)^2 * w(t) * t) / D_0) + sum(zeta,i,1,n)) ;
solx = solve(eqn, w(t))
This equation is now to be solved for w (t), but under the condition that v (t) is known.

Walter Roberson on 31 Jan 2021
syms rho_w w(t) p_u d_0 D_0 l h_0 t kappa m_St m_w rho_L c_w A_Q v(t) h_d lambda zeta n
syms sum_of_zeta
eqn = (rho_w / 2) * w(t)^2 + p_u == (rho_w / 2) * (d_0 / D_0)^4 * w(t)^2 + ((l - h_0) / (l - h_0 + (d_0 / D_0)^2 * w(t) * t))^kappa + (rho_w / (m_St + m_w - rho_w * (pi / 4) * d_0^2 * w(t) * t)) * (rho_w * (pi / 4) * d_0^2 * w(t)^2 - (rho_L / 2) * c_w * A_Q * v(t)^2) * (h_0 + h_d - (d_0 / D_0)^2 * w(t) * t) + (rho_w / 2) * (d_0 / D_0)^4 * w(t)^2 * (lambda * ((h_0 - (d_0 / D_0)^2 * w(t) * t) / D_0) + sum_of_zeta) ;
syms W V
eqnW = subs(eqn, w(t), W)
eqnW =
solw = solve(eqnW, W)
Warning: Unable to find explicit solution. For options, see help.
solw = Empty sym: 0-by-1
char(eqnW)
ans = 'p_u + (W^2*rho_w)/2 == (-(h_0 - l)/(l - h_0 + (W*d_0^2*t)/D_0^2))^kappa + (rho_w*((W^2*d_0^2*rho_w*pi)/4 - (A_Q*c_w*rho_L*v(t)^2)/2)*(h_0 + h_d - (W*d_0^2*t)/D_0^2))/(m_St + m_w - (W*d_0^2*rho_w*t*pi)/4) + (W^2*d_0^4*rho_w)/(2*D_0^4) + (W^2*d_0^4*rho_w*(sum_of_zeta + (lambda*(h_0 - (W*d_0^2*t)/D_0^2))/D_0))/(2*D_0^4)'
Walter Roberson on 1 Feb 2021
No, it is not possible for that equation to have 3 complex solutions and one real solution, not unless at least one of the coefficients are complex valued. Any polynomial system that has real-valued coefficients always has an even number of real-valued roots and the complex roots are always complex conjugates. Another other combination of roots multiplies out to have complex coefficients.

R2020b

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