# Proměnné a funkce t = var('t') x = function('x')(t)
# ODE ode = diff(x, t) == x - x^3
# Analytické řešení ODE s počáteční podmínkou x(0)=1/2 sol = desolve(ode, x, ics=[0, 1/2]) show(sol)
−12 log(x(t)+1)−12 log(x(t)−1)+log(x(t))=−12i π+t−log(2)−12 log(32)−12 log(12)\displaystyle -\frac{1}{2} \, \log\left(x\left(t\right) + 1\right) - \frac{1}{2} \, \log\left(x\left(t\right) - 1\right) + \log\left(x\left(t\right)\right) = -\frac{1}{2} i \, \pi + t - \log\left(2\right) - \frac{1}{2} \, \log\left(\frac{3}{2}\right) - \frac{1}{2} \, \log\left(\frac{1}{2}\right)−21log(x(t)+1)−21log(x(t)−1)+log(x(t))=−21iπ+t−log(2)−21log(23)−21log(21)
sol_exp = solve(sol, x) show(sol_exp)
[log(x(t))=−12i π+t−12 log(2)−12 log(32)+12 log(x(t)+1)+12 log(x(t)−1)]\displaystyle \left[\log\left(x\left(t\right)\right) = -\frac{1}{2} i \, \pi + t - \frac{1}{2} \, \log\left(2\right) - \frac{1}{2} \, \log\left(\frac{3}{2}\right) + \frac{1}{2} \, \log\left(x\left(t\right) + 1\right) + \frac{1}{2} \, \log\left(x\left(t\right) - 1\right)\right][log(x(t))=−21iπ+t−21log(2)−21log(23)+21log(x(t)+1)+21log(x(t)−1)]
# "Ruční" separace proměnných a integrace x_var = var('x') integrand = 1/(x_var - x_var^3) parfrac = integrand.partial_fraction(x_var) show(parfrac) integral_x = integrate(parfrac, x_var) show(integral_x) integral_t = integrate(1, t) show(integral_t) c = var('c') rce = integral_x == t + c show(rce)
−12 (x+1)−12 (x−1)+1x\displaystyle -\frac{1}{2 \, {\left(x + 1\right)}} - \frac{1}{2 \, {\left(x - 1\right)}} + \frac{1}{x}−2(x+1)1−2(x−1)1+x1
−12 log(x+1)−12 log(x−1)+log(x)\displaystyle -\frac{1}{2} \, \log\left(x + 1\right) - \frac{1}{2} \, \log\left(x - 1\right) + \log\left(x\right)−21log(x+1)−21log(x−1)+log(x)
t\displaystyle tt
−12 log(x+1)−12 log(x−1)+log(x)=c+t\displaystyle -\frac{1}{2} \, \log\left(x + 1\right) - \frac{1}{2} \, \log\left(x - 1\right) + \log\left(x\right) = c + t−21log(x+1)−21log(x−1)+log(x)=c+t
str(rce)
eq=-1/2*log(abs(x + 1)) - 1/2*log(abs(x - 1)) + log(abs(x)) == c + t
eq_sub = eq.subs({x: 1/2, t: 0})
show(eq_sub)
−12 log(32)+12 log(12)=c\displaystyle -\frac{1}{2} \, \log\left(\frac{3}{2}\right) + \frac{1}{2} \, \log\left(\frac{1}{2}\right) = c−21log(23)+21log(21)=c
c_val = solve(eq_sub, c)[0].rhs()
eq_c = eq.subs({c: c_val}) show(eq_c)
−12 log(∣x+1∣)−12 log(∣x−1∣)+log(∣x∣)=t−12 log(2)−12 log(32)\displaystyle -\frac{1}{2} \, \log\left({\left| x + 1 \right|}\right) - \frac{1}{2} \, \log\left({\left| x - 1 \right|}\right) + \log\left({\left| x \right|}\right) = t - \frac{1}{2} \, \log\left(2\right) - \frac{1}{2} \, \log\left(\frac{3}{2}\right)−21log(∣x+1∣)−21log(∣x−1∣)+log(∣x∣)=t−21log(2)−21log(23)
