Solving Separable Differential Equations
John Travis
Mississippi College
Blanchard, Devaney and Hall, 4th edition.
Enter the right-hand-side for a separable differential equation of the form
$\frac{\mathrm{dy}}{\mathrm{dt}} = f(t,y)$
with y as a function of t entered as "y(t)" instead of just plain "y".
Of course, the DE will be separable if it is possible to write $f(t,y) = g(y) h(t)$ for some functions $g$ and $h$.
# Worksheet still under development. Use at your own risk! 8-o
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var('t,k,M,m')
y = function('y', t)
f = 3*y^2/t
# f = 1*y*(10-y)*(y-2)
F = maxima.factor(f)
html('Differential Equation can be separated into\n\n<center>$dy/dx = %s$</center>'%str(latex(F)))
# Enter the domain value for the initial condition
t0=0
t1=2
# Enter the range value for the initial condition
y0=1
sol = desolve(diff(y,t) - f, y)
sol_p = desolve(diff(y,t) - f, y, ics=[t0,y0])
html('The exact general solution for this problem is given by $y(t)=%s $'%str(latex(sol)))
var('y')
G = plot(sol_p,(t,t0,t1)) + point2d((t0,y0),color='red',pointsize=20)
show(G)
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Differential Equation can be separated into
The exact general solution for this problem is given by verbose 0 (2387: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 200 points. verbose 0 (2387: plot.py, generate_plot_points) Last error message: 'unable to simplify to float approximation' 
Differential Equation can be separated into
The exact general solution for this problem is given by verbose 0 (2387: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 200 points. verbose 0 (2387: plot.py, generate_plot_points) Last error message: 'unable to simplify to float approximation'
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var('x,y,M')
f = 1/(x*(M-x))
f.show()
f.partial_fraction(x).show()
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# checking a nice example
var ('C')
f = integrate(x*sin(x),x)+C
soln = f/x^2
print soln
solnp = diff(soln,x)
print solnp
# and plug both back into original de
# x y' + 2 y = sin(x)
print
left = x*solnp +2*soln
left.show()
print left.simplify()
-(x*cos(x) - C - sin(x))/x^2 sin(x)/x + 2*(x*cos(x) - C - sin(x))/x^3 x*(sin(x)/x + 2*(x*cos(x) - C - sin(x))/x^3) - 2*(x*cos(x) - C - sin(x))/x^2 -(x*cos(x) - C - sin(x))/x^2 sin(x)/x + 2*(x*cos(x) - C - sin(x))/x^3 x*(sin(x)/x + 2*(x*cos(x) - C - sin(x))/x^3) - 2*(x*cos(x) - C - sin(x))/x^2 |
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