LAC09a Power Series MrG 2012.0612

3128 days ago by lac2012

#1) s'=s, s(0)=1 var('t') s=function('s',t) show(desolve(diff(s,t,1)-s,s)) show(desolve(diff(s,t,1)-s,s,[0,1])) show(taylor(e^t,t,0,5)) plot(desolve(diff(s,t,1)-s,s,[0,1])) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}c e^{t}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{t}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} \, t^{5} + \frac{1}{24} \, t^{4} + \frac{1}{6} \, t^{3} + \frac{1}{2} \, t^{2} + t + 1
\newcommand{\Bold}[1]{\mathbf{#1}}c e^{t}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{t}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} \, t^{5} + \frac{1}{24} \, t^{4} + \frac{1}{6} \, t^{3} + \frac{1}{2} \, t^{2} + t + 1
#2) s'=-s, s(0)=1 show(desolve(diff(s,t,1)+s,s)) show(desolve(diff(s,t,1)+s,s,[0,1])) show(taylor(e^(-t),t,0,5)) plot(desolve(diff(s,t,1)+s,s,[0,1])) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}c e^{\left(-t\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(-t\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{120} \, t^{5} + \frac{1}{24} \, t^{4} - \frac{1}{6} \, t^{3} + \frac{1}{2} \, t^{2} - t + 1
\newcommand{\Bold}[1]{\mathbf{#1}}c e^{\left(-t\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(-t\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{120} \, t^{5} + \frac{1}{24} \, t^{4} - \frac{1}{6} \, t^{3} + \frac{1}{2} \, t^{2} - t + 1
#3) s'=1/(1+t), s(0)=0 show(desolve(diff(s,t,1)-1/(1+t),s)) show(desolve(diff(s,t,1)-1/(1+t),s,[0,0])) show(taylor(log(abs(1+t)),t,0,5)) plot(desolve(diff(s,t,1)-1/(1+t),s,[0,0])) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}c + \log\left(t + 1\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(t + 1\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{5} \, t^{5} - \frac{1}{4} \, t^{4} + \frac{1}{3} \, t^{3} - \frac{1}{2} \, t^{2} + t
\newcommand{\Bold}[1]{\mathbf{#1}}c + \log\left(t + 1\right)
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\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{5} \, t^{5} - \frac{1}{4} \, t^{4} + \frac{1}{3} \, t^{3} - \frac{1}{2} \, t^{2} + t
#4a) s''=-s, s(0)=1, s'(0)=0 show(desolve(diff(s,t,2)+s,s)) show(desolve(diff(s,t,2)+s,s,[0,1,0])) show(taylor(cos(t),t,0,6)) plot(desolve(diff(s,t,2)+s,s,[0,1,0]),0,2*pi) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}k_{1} \sin\left(t\right) + k_{2} \cos\left(t\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(t\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{720} \, t^{6} + \frac{1}{24} \, t^{4} - \frac{1}{2} \, t^{2} + 1
\newcommand{\Bold}[1]{\mathbf{#1}}k_{1} \sin\left(t\right) + k_{2} \cos\left(t\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(t\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{720} \, t^{6} + \frac{1}{24} \, t^{4} - \frac{1}{2} \, t^{2} + 1
#4b) s''=-s, s(0)=0, s'(0)=1 show(desolve(diff(s,t,2)+s,s)) show(desolve(diff(s,t,2)+s,s,[0,0,1])) show(taylor(sin(t),t,0,6)) plot(desolve(diff(s,t,2)+s,s,[0,0,1]),0,2*pi) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}k_{1} \sin\left(t\right) + k_{2} \cos\left(t\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(t\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} \, t^{5} - \frac{1}{6} \, t^{3} + t
\newcommand{\Bold}[1]{\mathbf{#1}}k_{1} \sin\left(t\right) + k_{2} \cos\left(t\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(t\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} \, t^{5} - \frac{1}{6} \, t^{3} + t
#5) s'=t+s, s(0)=0 show(desolve(diff(s,t,1)-t-s,s)) show(desolve(diff(s,t,1)-t-s,s,[0,0])) show(taylor(e^t-t-1,t,0,5)) plot(desolve(diff(s,t,1)-t-s,s,[0,0])) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-{\left({\left(t + 1\right)} e^{\left(-t\right)} - c\right)} e^{t}
\newcommand{\Bold}[1]{\mathbf{#1}}-t + e^{t} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} \, t^{5} + \frac{1}{24} \, t^{4} + \frac{1}{6} \, t^{3} + \frac{1}{2} \, t^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}-{\left({\left(t + 1\right)} e^{\left(-t\right)} - c\right)} e^{t}
\newcommand{\Bold}[1]{\mathbf{#1}}-t + e^{t} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{120} \, t^{5} + \frac{1}{24} \, t^{4} + \frac{1}{6} \, t^{3} + \frac{1}{2} \, t^{2}