solving equation with variables.

I wanna solve this equation for K_1^2/w^2, when A=0

A=-(E*G*k_1^4 + k_1^2*r^2*w^2 - E*R*k_1^2*w^2 - e*m*r^2*w^4 + E^2*R*l^2*m*w^4 + E*R*e*m*w^4 - E^2*G*k_1^2*l^2*m*w^2 - E*G*e*k_1^2*m*w^2)/(G*e)

Answer 1

[Star Strider]

May not be the most efficient code, but it seems to work:

syms E G R k_1 r w e m l a b 
assume( (k_1^2/w^2) > 0)
assumeAlso( k_1 > 0 )
assumeAlso( w > 0 )
% Equation: 
A=-(E*G*k_1^4 + k_1^2*r^2*w^2 - E*R*k_1^2*w^2 - e*m*r^2*w^4 + E^2*R*l^2*m*w^4 + E*R*e*m*w^4 - E^2*G*k_1^2*l^2*m*w^2 - E*G*e*k_1^2*m*w^2)/(G*e);

% Substitute ‘a’ for ‘k_1^2’ and ‘b’ for ‘w^2’ 
B = subs(A == 0, {k_1^2, w^2, k_1^4, w^4}, {a, b, a^2, b^2})

B = simplify(collect(expand(B)))

% Solve for ‘a’ = ‘k_1^2’
a = solve(B, a);
% Factor out ‘b’ = ‘w^2’ (not necessary but helpful)
a = collect(a, b);

% ‘c’ = a/b = k_1^2/w^2
c = simplify(collect(a/b))

Answer 2

[Walter Roberson]

syms E G R k_1 r w e m l a b B
A = -(E*G*k_1^4 + k_1^2*r^2*w^2 - E*R*k_1^2*w^2 - e*m*r^2*w^4 + E^2*R*l^2*m*w^4 + E*R*e*m*w^4 - E^2*G*k_1^2*l^2*m*w^2 - E*G*e*k_1^2*m*w^2)/(G*e);
Bsol = simplify( solve( subs(A, w, sqrt(B/k_1^2)), B) );

Bsol will be k_1^2 / w^2

there will be two solutions (i.e. quadratic form)