John Travis
Mississippi College
Playing around with solutions of various first-order differential equations.
html('Solving the first order linear DE $\\frac{dy}{dt} = P(t)y+Q(t)$ using Sage symbolics only')
var('t')
y = function('y',t)
P = function('P',t)
Q = function('Q',t)
DE = diff(y,t)==P*y+Q
soln=desolve(DE,y,ivar=t)
html('<p><font size=+1>Solution is $y(t) = %s'%str(latex(soln)))
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Solving the first order linear DE \frac{dy}{dt} = P(t)y+Q(t) using Sage symbolics only
Solution is y(t) = {\left(c + \int e^{\left(-\int P\left(t\right)\,{d t}\right)} Q\left(t\right)\,{d t}\right)} e^{\left(\int P\left(t\right)\,{d t}\right)}
Solving the first order linear DE \frac{dy}{dt} = P(t)y+Q(t) using Sage symbolics only
Solution is y(t) = {\left(c + \int e^{\left(-\int P\left(t\right)\,{d t}\right)} Q\left(t\right)\,{d t}\right)} e^{\left(\int P\left(t\right)\,{d t}\right)} |
# Solving a particular first order linear DE using Sage
var('t')
y = function('y',t)
P = function('P',t)
Q = function('Q',t)
assume(t>2)
P = 1/sqrt(t^3-3)
Q = t
html('<font size=+1>Solving $\\frac{dy}{dt} = %s$'%str(latex(P))+'$y + %s$</font>'%str(latex(Q)))
DE = diff(y,t)==P*y+Q
soln=desolve(DE,y,ivar=t,ics=[1,3])
html('<p><font size=+1>Solution is $y(t) = %s'%str(latex(soln.expand())))
G=points(desolve_rk4(DE, y, ics=[1,3], ivar=t, end_points=3, step=0.1))
G+=plot(soln,t,[1,4])
show(G)
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Solving \frac{dy}{dt} = \frac{1}{\sqrt{t^{3} - 3}}y + t
Traceback (click to the left of this block for traceback) ... ValueError: The name "y(t)" is not a valid Python identifier.
Solving \frac{dy}{dt} = \frac{1}{\sqrt{t^{3} - 3}}y + t
Traceback (most recent call last): P = 1/sqrt(t^3-3)
File "", line 1, in <module>
File "/tmp/tmpSF0IHV/___code___.py", line 13, in <module>
soln=desolve(DE,y,ivar=t,ics=[_sage_const_1 ,_sage_const_3 ])
File "/usr/local/sage-5.3/local/lib/python2.7/site-packages/sage/calculus/desolvers.py", line 501, in desolve
soln=soln.sage()
File "/usr/local/sage-5.3/local/lib/python2.7/site-packages/sage/interfaces/interface.py", line 865, in sage
return self._sage_()
File "/usr/local/sage-5.3/local/lib/python2.7/site-packages/sage/interfaces/maxima_abstract.py", line 1224, in _sage_
maxima=self.parent())
File "/usr/local/sage-5.3/local/lib/python2.7/site-packages/sage/calculus/calculus.py", line 1789, in symbolic_expression_from_maxima_string
return symbolic_expression_from_string(s, syms, accept_sequence=True)
File "/usr/local/sage-5.3/local/lib/python2.7/site-packages/sage/calculus/calculus.py", line 1961, in symbolic_expression_from_string
return parse_func(s)
File "parser.pyx", line 516, in sage.misc.parser.Parser.parse_sequence (sage/misc/parser.c:4109)
File "parser.pyx", line 531, in sage.misc.parser.Parser.parse_sequence (sage/misc/parser.c:3990)
File "parser.pyx", line 602, in sage.misc.parser.Parser.p_sequence (sage/misc/parser.c:4731)
File "parser.pyx", line 692, in sage.misc.parser.Parser.p_eqn (sage/misc/parser.c:5513)
File "parser.pyx", line 728, in sage.misc.parser.Parser.p_expr (sage/misc/parser.c:5816)
File "parser.pyx", line 761, in sage.misc.parser.Parser.p_term (sage/misc/parser.c:6056)
File "parser.pyx", line 800, in sage.misc.parser.Parser.p_factor (sage/misc/parser.c:6410)
File "parser.pyx", line 803, in sage.misc.parser.Parser.p_factor (sage/misc/parser.c:6439)
File "parser.pyx", line 833, in sage.misc.parser.Parser.p_power (sage/misc/parser.c:6599)
File "parser.pyx", line 803, in sage.misc.parser.Parser.p_factor (sage/misc/parser.c:6439)
File "parser.pyx", line 830, in sage.misc.parser.Parser.p_power (sage/misc/parser.c:6568)
File "parser.pyx", line 893, in sage.misc.parser.Parser.p_atom (sage/misc/parser.c:7203)
File "parser.pyx", line 731, in sage.misc.parser.Parser.p_expr (sage/misc/parser.c:5854)
File "parser.pyx", line 761, in sage.misc.parser.Parser.p_term (sage/misc/parser.c:6056)
File "parser.pyx", line 803, in sage.misc.parser.Parser.p_factor (sage/misc/parser.c:6439)
File "parser.pyx", line 830, in sage.misc.parser.Parser.p_power (sage/misc/parser.c:6568)
File "parser.pyx", line 888, in sage.misc.parser.Parser.p_atom (sage/misc/parser.c:7142)
File "/usr/local/sage-5.3/local/lib/python2.7/site-packages/sage/calculus/calculus.py", line 1451, in at
return ex.subs(**kwds)
File "expression.pyx", line 3753, in sage.symbolic.expression.Expression.substitute (sage/symbolic/expression.cpp:18208)
File "ring.pyx", line 520, in sage.symbolic.ring.SymbolicRing.var (sage/symbolic/ring.cpp:6956)
File "ring.pyx", line 568, in sage.symbolic.ring.SymbolicRing.var (sage/symbolic/ring.cpp:6689)
ValueError: The name "y(t)" is not a valid Python identifier.
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html('Solving a Bernoulli DE $\\frac{dy}{dt} = P(t)y+Q(t)y^2$ using Sage symbolics only')
var('t')
y = function('y',t)
P = function('P',t)
Q = function('Q',t)
DE = diff(y,t)==P*y+Q*y^2
soln=desolve(DE,y,ivar=t)
html('<p><font size=+1>Solution is $y(t) = %s'%str(latex(soln)))
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Solving a Bernoulli DE \frac{dy}{dt} = P(t)y+Q(t)y^2 using Sage symbolics only
Solution is y(t) = \frac{e^{\left(\int P\left(t\right)\,{d t}\right)}}{c - \int e^{\left(\int P\left(t\right)\,{d t}\right)} Q\left(t\right)\,{d t}}
Solving a Bernoulli DE \frac{dy}{dt} = P(t)y+Q(t)y^2 using Sage symbolics only
Solution is y(t) = \frac{e^{\left(\int P\left(t\right)\,{d t}\right)}}{c - \int e^{\left(\int P\left(t\right)\,{d t}\right)} Q\left(t\right)\,{d t}} |
# Solving a particular Bernoulli DE using Sage
var('t')
y = function('y',t)
P = function('P',t)
Q = function('Q',t)
P = 2/t
Q = -t^2
html('<font size=+1>Solving $\\frac{dy}{dt} = %s$'%str(latex(P))+'$y + %sy^2$</font>'%str(latex(Q)))
DE = diff(y,t)==P*y+Q*y^2
soln=desolve(DE,y,ivar=t,ics=[1,3])
html('<font size=+1>Solution is $y(t) = %s'%str(latex(soln.expand())))
G=plot(soln,t,[1,4])
#G+=points(desolve_rk4(DE, y, ics=[1,3], ivar=t, end_points=3, step=0.1))
show(G)
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Solving \frac{dy}{dt} = \frac{2}{t}y + -t^{2}y^2
Solution is y(t) = \frac{15 \, t^{2}}{3 \, t^{5} + 2} 
Solving \frac{dy}{dt} = \frac{2}{t}y + -t^{2}y^2
Solution is y(t) = \frac{15 \, t^{2}}{3 \, t^{5} + 2}
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# what about a DE with an arbitrary power on the last y...
# notice, n=0 is first order linear...does solution fit? Yes
# notice, n=1 is separable...does solution fit? No
html('Solving $\\frac{dy}{dt} = P(t)y+Q(t)y^n$ using Sage symbolics only')
var('t,n')
y = function('y',t)
P = function('P',t)
Q = function('Q',t)
DE = diff(y,t)==P*y+Q*y^n
soln=desolve(DE,y,ivar=t)
html('<p><font size=+1>Solution is $y(t) = %s'%str(latex(soln)))
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Solving \frac{dy}{dt} = P(t)y+Q(t)y^n using Sage symbolics only
Solution is y(t) = {\left(-{\left(n - 1\right)} \int e^{\left({\left(n - 1\right)} \int P\left(t\right)\,{d t}\right)} Q\left(t\right)\,{d t} + c\right)}^{{\left(-\frac{1}{n - 1}\right)}} e^{\left(\int P\left(t\right)\,{d t}\right)}
Solving \frac{dy}{dt} = P(t)y+Q(t)y^n using Sage symbolics only
Solution is y(t) = {\left(-{\left(n - 1\right)} \int e^{\left({\left(n - 1\right)} \int P\left(t\right)\,{d t}\right)} Q\left(t\right)\,{d t} + c\right)}^{{\left(-\frac{1}{n - 1}\right)}} e^{\left(\int P\left(t\right)\,{d t}\right)} |
# Solving a particular wierd DE using Sage
var('t')
y = function('y',t)
P = function('P',t)
Q = function('Q',t)
P = 1/t^2
Q = 4
n=5
html('<font size=+1>Solving $\\frac{dy}{dt} = %s$'%str(latex(P))+'$y + %sy^n$</font>'%str(latex(Q)))
DE = diff(y,t)==P*y+Q*y^n
soln=desolve(DE,y,ivar=t,ics=[1,3])
html('<font size=+1>Solution is $y(t) = %s'%str(latex(soln.expand())))
G=plot(soln,t,[1,4])
#G+=points(desolve_rk4(DE, y, ics=[1,3], ivar=t, end_points=3, step=0.1))
show(G)
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Solving \frac{dy}{dt} = \frac{1}{t^{2}}y + 4y^n
Solution is y(t) = \frac{3 \, e^{\left(-\frac{1}{t} + 1\right)}}{{\left(-5184 \, e^{4} \Gamma\left(-1, \frac{4}{t}\right) + 5184 \, e^{4} \Gamma\left(-1, 4\right) + 1\right)}^{\frac{1}{4}}} verbose 0 (4075: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 199 points. verbose 0 (4075: plot.py, generate_plot_points) Last error message: 'negative number to a fractional power not real' 
Solving \frac{dy}{dt} = \frac{1}{t^{2}}y + 4y^n
Solution is y(t) = \frac{3 \, e^{\left(-\frac{1}{t} + 1\right)}}{{\left(-5184 \, e^{4} \Gamma\left(-1, \frac{4}{t}\right) + 5184 \, e^{4} \Gamma\left(-1, 4\right) + 1\right)}^{\frac{1}{4}}} verbose 0 (4075: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 199 points. verbose 0 (4075: plot.py, generate_plot_points) Last error message: 'negative number to a fractional power not real'
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