LAC10 Friction 2012.0613 -- Sage

#1) hyperbolic functions a=plot(sinh(x),-2,2,color='red') b=plot(cosh(x),-2,2,color='green') c=plot(tanh(x),-2,2,color='cyan') plot(a+b+c,aspect_ratio=1)

plot([(e^x-e^(-x))/2,(e^x+e^(-x))/2,(e^x-e^(-x))/(e^x+e^(-x))],-2,2,aspect_ratio=1)

#2) hyperbolic derivatives show(diff(sinh(x),x)) show(diff(cosh(x),x)) show(diff(tanh(x),x))

\newcommand{\Bold}[1]{\mathbf{#1}}\cosh\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sinh\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\tanh\left(x\right)^{2} + 1

\newcommand{\Bold}[1]{\mathbf{#1}}\cosh\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sinh\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\tanh\left(x\right)^{2} + 1

#3) hyperbolic antiderivatives show(integrate(sinh(x),x)) show(integrate(cosh(x),x)) show(integrate(tanh(x),x))

\newcommand{\Bold}[1]{\mathbf{#1}}\cosh\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sinh\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\cosh\left(x\right)\right)

\newcommand{\Bold}[1]{\mathbf{#1}}\cosh\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sinh\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\cosh\left(x\right)\right)

#4) v'=g-(k/m)v^2, v(0)=0 var('t') v=function('v',t) show(desolve(diff(v,t,1)-32+3/5*v^2,v)) show(desolve(diff(v,t,1)-32+3/5*v^2,v,[0,0])) show(solve(desolve(diff(v,t,1)-32+3/5*v^2,v,[0,0]),v)) l=solve(desolve(diff(v,t,1)-32+3/5*v^2,v,[0,0]),v) plot(l[0].rhs())

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{48} \, \sqrt{30} \log\left(-\frac{4 \, \sqrt{30} - 3 \, v\left(t\right)}{4 \, \sqrt{30} + 3 \, v\left(t\right)}\right) = c + t
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{48} \, \sqrt{30} \log\left(-\frac{4 \, \sqrt{30} - 3 \, v\left(t\right)}{4 \, \sqrt{30} + 3 \, v\left(t\right)}\right) = \frac{1}{240} \, {\left(-5 i \, \pi + 8 \, \sqrt{30} t\right)} \sqrt{30}
\newcommand{\Bold}[1]{\mathbf{#1}}\left[v\left(t\right) = \frac{4 \, {\left(\sqrt{30} e^{\left(\frac{8}{5} \, \sqrt{30} t\right)} - \sqrt{30}\right)}}{3 \, {\left(e^{\left(\frac{8}{5} \, \sqrt{30} t\right)} + 1\right)}}\right]

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{48} \, \sqrt{30} \log\left(-\frac{4 \, \sqrt{30} - 3 \, v\left(t\right)}{4 \, \sqrt{30} + 3 \, v\left(t\right)}\right) = c + t
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{48} \, \sqrt{30} \log\left(-\frac{4 \, \sqrt{30} - 3 \, v\left(t\right)}{4 \, \sqrt{30} + 3 \, v\left(t\right)}\right) = \frac{1}{240} \, {\left(-5 i \, \pi + 8 \, \sqrt{30} t\right)} \sqrt{30}
\newcommand{\Bold}[1]{\mathbf{#1}}\left[v\left(t\right) = \frac{4 \, {\left(\sqrt{30} e^{\left(\frac{8}{5} \, \sqrt{30} t\right)} - \sqrt{30}\right)}}{3 \, {\left(e^{\left(\frac{8}{5} \, \sqrt{30} t\right)} + 1\right)}}\right]

#5) y'=(e^t-1)/(e^t+1), y(0)=0 y=function('y',t) show(desolve(diff(y,t,1)-(e^t-1)/(e^t+1),y)) show(desolve(diff(y,t,1)-(e^t-1)/(e^t+1),y,[0,0])) show(desolve(diff(y,t,1)-(e^t-1)/(e^t+1),y,[0,0]).simplify_log()) plot(desolve(diff(y,t,1)-(e^t-1)/(e^t+1),y,[0,0]).simplify_log())

\newcommand{\Bold}[1]{\mathbf{#1}}c - t + 2 \, \log\left(e^{t} + 1\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-t + 2 \, \log\left(e^{t} + 1\right) - 2 \, \log\left(2\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-t + \log\left(\frac{1}{4} \, {\left(e^{t} + 1\right)}^{2}\right)

\newcommand{\Bold}[1]{\mathbf{#1}}c - t + 2 \, \log\left(e^{t} + 1\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-t + 2 \, \log\left(e^{t} + 1\right) - 2 \, \log\left(2\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-t + \log\left(\frac{1}{4} \, {\left(e^{t} + 1\right)}^{2}\right)

#6) y'=tanh(t), y(0)=0 show(desolve(diff(y,t,1)-tanh(t),y)) show(desolve(diff(y,t,1)-tanh(t),y,[0,0])) plot(desolve(diff(y,t,1)-tanh(t),y,[0,0]))

\newcommand{\Bold}[1]{\mathbf{#1}}c + \log\left(\cosh\left(t\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\cosh\left(t\right)\right)

\newcommand{\Bold}[1]{\mathbf{#1}}c + \log\left(\cosh\left(t\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\cosh\left(t\right)\right)

LAC10 Friction 2012.0613

3299 days ago by lac2012