Book
Advanced Engineering Mathematics
Author
Erwin Kreyszig
Edition
8th Edition
<p><strong>section 1.4 kreyszig, </strong></p>
<p><strong>problem 1 </strong></p>
<p>$y' \propto y \Rightarrow y'(t) = k*y(t)$ </p>
<p>we solve the differential equation using sage..</p>
{{{id=1|
t = var('t') # define a variable t
k = var('k')
y0 = var('y0')
y = function('y',t) # define x to be a function of that variable
DE = diff(y, t) -k*y
desolve(DE, [y,t],ics=(0,y0))
///
y0*e^(k*t)<p>it has been given that the initial value y0 is doubled after 1 day</p> <p>$\Rightarrow y(1) = 2*y0*e^{k} \Rightarrow k = \ln{2}$</p> <p>$\Rightarrow y(t) = 2*y0*e^{t}$</p>
<p> </p> <div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow-x: hidden; overflow-y: hidden;"><p>it has been given that the initial value y0 is doubled after 1 day</p></div> <div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow-x: hidden; overflow-y: hidden;"><p>$\Rightarrow y(1) = 2*y0*e^{k} \Rightarrow k = \ln{2}mce_markerlt;/p></div> <div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow-x: hidden; overflow-y: hidden;"><p>$\Rightarrow y(t) = 2*y0*e^{t}mce_markerlt;/p></div> <div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow-x: hidden; overflow-y: hidden;"><p>(i) after 3 days, y(3) = 2*e^{3}*y0</p></div> <div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow-x: hidden; overflow-y: hidden;"><p>(ii) (i) after a week , y(3) = 2*e^{7}*y0</p></div> <p> </p> <p>it has been given that the initial value y0 is doubled after 1 day</p> <p>$\Rightarrow y(1) = 2*y0*e^{k} \Rightarrow k = \ln{2}$</p> <p>$\Rightarrow y(t) = 2*y0*e^{t}$</p> <p>(i) after 3 days, $y(3) = 2*e^{3}*y0$</p> <p>(ii) after a week , $y(3) = 2*e^{7}*y0$</p> <p> </p> <p> </p>
<p>taking</p> <p>$y0 = 1$,</p> <p>we plot here the function</p> <p>$y(t)$</p>
{{{id=5| t = var('t') y(t) = 2*exp(t) plot(y,t) /// <html><font color='black'><img src='cell://sage0.png'></font></html> }}}
{{{id=6|
/// }}}
{{{id=8|
/// }}}
<p><strong><br /></strong></p>
{{{id=9|
/// }}}
}}}
Solution by:
- ganesh, developer, SPACE
- bhanukiran, student, Technical university DELFT
- saransh, student, shobit university meerut
- ujwala Dhake, teacher, Aurangabad