Exàmen Pràctiques 17-01-17.sagews

# Exercici 2
t=var("t")
x=function("x",t)
y=function("y",t)
eqx=diff(x,t)==3+(6*y(t)/100)-(14*x(t))/100
eqy=diff(y,t)==3+(4*x(t)/100)-(12*y(t))/100
solucio=desolve_system([eqx,eqy],[x,y],[0,0,0])
show(solucio)
expand(solucio[1](10))

[$\displaystyle x\left(t\right) = -\frac{75}{2} \, e^{\left(-\frac{2}{25} \, t\right)} + \frac{75}{2}$, $\displaystyle y\left(t\right) = -\frac{75}{2} \, e^{\left(-\frac{2}{25} \, t\right)} + \frac{75}{2}$]

y(10) == -75/2*e^(-4/5) + 75/2

n(-75/2*e^(-4/5) + 75/2)
limit(solucio[0],t=infinity)

20.6501638456042 limit(x(t), t, +Infinity) == (75/2)

# Exercici 1
t=var("t")
r=var("r")
l=var("l")
c=var("c")
p=var("p")
v=var("v")
a=function("a",t)
i=function("i",t)
i=p*exp(t)*cos(5*t+v)
i1=diff(i,t)
i2=diff(i1,t)
eq=0==i1+i2+i/c
sol=solve(eq,c)
c=function("c",t)
c=(cos(5*t+v))/(23*cos(5*t+v)+15*sin(5*t+v))
q=function("q",t)
r=function("r",t)
equacio=13==diff(q,t)+diff(diff(q,t))+c*q
eqa=diff(q,t)==r(t)
eqb=diff(r,t)==13-r(t)-c*q
show(eqa,eqb)
sols=desolve_system([eqa,eqb], [q,r], [0,1,3])

$\displaystyle D[0]\left(q\right)\left(t\right) = r\left(t\right)$ $\displaystyle D[0]\left(r\right)\left(t\right) = -\frac{\cos\left(5 \, t + v\right) q\left(t\right)}{23 \, \cos\left(5 \, t + v\right) + 15 \, \sin\left(5 \, t + v\right)} - r\left(t\right) + 13$

Error in lines 22-22 Traceback (most recent call last): File "/projects/sage/sage-7.3/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 976, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-7.3/local/lib/python2.7/site-packages/sage/calculus/desolvers.py", line 817, in desolve_system raise ValueError("Unable to determine independent variable, please specify.") ValueError: Unable to determine independent variable, please specify.

b=function("b",t)
c=var("c")
assume(0<c<1)
eq1=-8*pi*sin(8*pi*t)+12*pi*cos(12*pi*t)==diff(a,t)+2*diff(diff(a,t))+a(t)/c
eq1homo=0==diff(a,t)+2*diff(diff(a,t))+a(t)/c
solucio3=desolve(eq1homo,[a,t]);
show(solucio3)

$\displaystyle {\left(K_{2} \cos\left(\frac{1}{2} \, t \sqrt{\frac{2}{c} - \frac{1}{4}}\right) + K_{1} \sin\left(\frac{1}{2} \, t \sqrt{\frac{2}{c} - \frac{1}{4}}\right)\right)} e^{\left(-\frac{1}{4} \, t\right)}$

limit(solucio3,t=infinity)

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