LAC03-05 DiffEqus MrG 2012.0601

3118 days ago by lac2012

#1a) VARSEP: dy/dt=0.2y var('y') l=sqrt(1+0.04*y^2) p1a=plot_vector_field((1/l,0.2*y/l),(x,0,10),(y,0,23)) show(p1a) 
       
#2a) VARSEP: dy/dt=0.2(100-y) l=sqrt(1+0.04*(100-y)^2) p2a=plot_vector_field((1/l,0.2*(100-y)/l),(x,0,50),(y,0,200)) show(p2a) 
       
#3a) VARSEP: dy/dt=0.01y*(100-y) l=sqrt(1+0.0001*y^2*(100-y)^2) p3a=plot_vector_field((1/l,0.01*y*(100-y)/l),(x,0,10),(y,0,200)) show(p3a) 
       
#1b) VARSEP: y'=0.2y, y(0)=3 y=function('y',x) show(desolve(diff(y,x,1)-0.2*y,y)) show(desolve(diff(y,x,1)-0.2*y,y,[0,3])) p1b=plot(desolve(diff(y,x,1)-0.2*y,y,[0,3]),xmin=0,xmax=10) show(p1a+p1b) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}c e^{\left(\frac{1}{5} \, x\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, e^{\left(\frac{1}{5} \, x\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}c e^{\left(\frac{1}{5} \, x\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, e^{\left(\frac{1}{5} \, x\right)}
#2b) VARSEP: y'=0.2(100-y),y(0)=10 show(desolve(diff(y,x,1)-0.2*(100-y),y)) show(desolve(diff(y,x,1)-0.2*(100-y),y,[0,10])) p2b=plot(desolve(diff(y,x,1)-0.2*(100-y),y,[0,10]),xmin=0,xmax=50,ymin=0,ymax=200) show(p2a+p2b) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(c + 100 \, e^{\left(\frac{1}{5} \, x\right)}\right)} e^{\left(-\frac{1}{5} \, x\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}10 \, {\left(10 \, e^{\left(\frac{1}{5} \, x\right)} - 9\right)} e^{\left(-\frac{1}{5} \, x\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(c + 100 \, e^{\left(\frac{1}{5} \, x\right)}\right)} e^{\left(-\frac{1}{5} \, x\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}10 \, {\left(10 \, e^{\left(\frac{1}{5} \, x\right)} - 9\right)} e^{\left(-\frac{1}{5} \, x\right)}
#3b) VARSEP: dy/dt=0.01y*(100-y), y(0)=200 show(desolve(diff(y,x,1)-0.01*y*(100-y),y)) show(desolve(diff(y,x,1)-0.01*y*(100-y),y,[0,200])) show(desolve(diff(y,x,1)-0.01*y*(100-y),y,[0,200]).simplify_radical().simplify_log()) equ=solve(desolve(diff(y,x,1)-0.01*y*(100-y),y,[0,200]).simplify_radical().simplify_log(),y) show(equ) p3b=plot(200*e^x/(2*e^x-1),xmin=0,xmax=10,ymin=0,ymax=200) show(p3a+p3b) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-\log\left(y\left(x\right) - 100\right) + \log\left(y\left(x\right)\right) = c + x
\newcommand{\Bold}[1]{\mathbf{#1}}-\log\left(y\left(x\right) - 100\right) + \log\left(y\left(x\right)\right) = x - \log\left(100\right) + \log\left(200\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\frac{y\left(x\right)}{y\left(x\right) - 100}\right) = x + \log\left(2\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y\left(x\right) = \frac{200 \, e^{x}}{2 \, e^{x} - 1}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}-\log\left(y\left(x\right) - 100\right) + \log\left(y\left(x\right)\right) = c + x
\newcommand{\Bold}[1]{\mathbf{#1}}-\log\left(y\left(x\right) - 100\right) + \log\left(y\left(x\right)\right) = x - \log\left(100\right) + \log\left(200\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\frac{y\left(x\right)}{y\left(x\right) - 100}\right) = x + \log\left(2\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y\left(x\right) = \frac{200 \, e^{x}}{2 \, e^{x} - 1}\right]
#4a) UAM: Galilean Free Fall Model, y"=-g, v'=-g var('t') y=function('y',t) show(desolve(diff(y,t,2)+32,y)) show(desolve(diff(y,t,2)+32,y,[0,10,100])) p1=plot(desolve(diff(y,t,2)+32,y,[0,10,100]),0,2*100/32,color='red') v=function('v',t) show(desolve(diff(v,t,1)+32,v)) show(desolve(diff(v,t,1)+32,v,[0,100])) p2=plot(desolve(diff(v,t,1)+32,v,[0,100]),0,2*100/32,color='green') show(p1+p2) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}k_{2} t - 16 \, t^{2} + k_{1}
\newcommand{\Bold}[1]{\mathbf{#1}}-16 \, t^{2} + 100 \, t + 10
\newcommand{\Bold}[1]{\mathbf{#1}}c - 32 \, t
\newcommand{\Bold}[1]{\mathbf{#1}}-32 \, t + 100
\newcommand{\Bold}[1]{\mathbf{#1}}k_{2} t - 16 \, t^{2} + k_{1}
\newcommand{\Bold}[1]{\mathbf{#1}}-16 \, t^{2} + 100 \, t + 10
\newcommand{\Bold}[1]{\mathbf{#1}}c - 32 \, t
\newcommand{\Bold}[1]{\mathbf{#1}}-32 \, t + 100