In [24]:

    

from sympy import *
init_printing()
var('t,mu,omega,omega0,F0,A,B')
x=A*cos(omega0*t)+B*sin(omega0*t)
eq=x.diff(t,2)+2*mu*x.diff(t)+omega**2*x-F0*cos(omega0*t)
eqL1=eq.factor().coeff(sin(omega0*t))
eqL2=eq.factor().coeff(cos(omega0*t))
Incog=[A,B]
Ecuas=[eqL1,eqL2]
M=Matrix([[ecu.coeff(inco) for inco in Incog] for ecu in Ecuas])
print(latex(M.det()))


    

    




- 4 \mu^{2} \omega_{0}^{2} - \omega^{4} + 2 \omega^{2} \omega_{0}^{2} - \omega_{0}^{4}

In [26]:

    

print(latex(solve(M.det(),omega)))


    

    




\left [ - \sqrt{- 2 i \mu \omega_{0} + \omega_{0}^{2}}, \quad \sqrt{- 2 i \mu \omega_{0} + \omega_{0}^{2}}, \quad - \sqrt{2 i \mu \omega_{0} + \omega_{0}^{2}}, \quad \sqrt{2 i \mu \omega_{0} + \omega_{0}^{2}}\right ]

In [29]:

    

SolAB=solve([eqL1,eqL2],[A,B])  
x.subs(SolAB)


    

    
Out[29]:





$$\frac{2 F_{0} \mu \omega_{0} \sin{\left (\omega_{0} t \right )}}{4 \mu^{2} \omega_{0}^{2} + \left(\omega^{2} - \omega_{0}^{2}\right)^{2}} + \frac{F_{0} \left(\omega^{2} - \omega_{0}^{2}\right) \cos{\left (\omega_{0} t \right )}}{4 \mu^{2} \omega_{0}^{2} + \left(\omega^{2} - \omega_{0}^{2}\right)^{2}}$$

In [32]:

    

rho=sqrt(A**2+B**2).subs(SolAB).simplify()
rho


    

    
Out[32]:





$$\sqrt{\frac{F_{0}^{2}}{4 \mu^{2} \omega_{0}^{2} + \left(\omega^{2} - \omega_{0}^{2}\right)^{2}}}$$

In [36]:

    

plot(rho.subs({F0:1, mu:.1,omega:5}),(omega0,0,10) )


    

    
Out[36]:





<sympy.plotting.plot.Plot at 0x7fd4d05bb490>

In [57]:

    

sol=solve(rho.diff(omega0),omega0)
sol


    

    
Out[57]:





$$\left [ 0, \quad - \sqrt{- 2 \mu^{2} + \omega^{2}}, \quad \sqrt{- 2 \mu^{2} + \omega^{2}}, \quad - \sqrt{\tilde{\infty} \sqrt{F_{0}^{2}} - 2 \mu^{2} + \omega^{2}}, \quad \sqrt{\tilde{\infty} \sqrt{F_{0}^{2}} - 2 \mu^{2} + \omega^{2}}\right ]$$

In [61]:

    

rho.diff(omega0,2).subs(omega0,sol[2])


    

    
Out[61]:





$$\frac{2 \sqrt{\frac{F_{0}^{2}}{4 \mu^{4} + 4 \mu^{2} \left(- 2 \mu^{2} + \omega^{2}\right)}} \left(4 \mu^{2} - 2 \omega^{2}\right)}{4 \mu^{4} + 4 \mu^{2} \left(- 2 \mu^{2} + \omega^{2}\right)}$$

In [63]:

    

ho.subs(omega0,sol[2]).simplify()


    

    
Out[63]:





$$\frac{1}{2} \sqrt{\frac{F_{0}^{2}}{\mu^{2} \left(- \mu^{2} + \omega^{2}\right)}}$$

In [64]:

    

rho.subs(omega0,0).simplify()


    

    
Out[64]:





$$\sqrt{\frac{F_{0}^{2}}{\omega^{4}}}$$

In [67]:

    

limit(rho,omega0,oo)


    

    
Out[67]:





$$0$$

In [69]:

    

sol[2].subs({mu:.1,omega:5})


    

    
Out[69]:





$$4.99799959983992$$

In [ ]: