ode1.html

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#----------------------------------------------------
# The ODE (second order correction for weakly damped
# mass spring damper system)
#----------------------------------------------------
> ode:=diff(x(t),t$2)+x(t)=-1/2*t*sin(t);

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#----------------------------------------------------
# ...and its homogeneous counterpart
#----------------------------------------------------
> ode_h:=diff(x(t),t$2)+x(t);

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#----------------------------------------------------
# Solve the bloody thing using maple
#----------------------------------------------------
> dsolve(ode,x(t));

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#----------------------------------------------------
# Now do it "by hand".
#----------------------------------------------------
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#----------------------------------------------------
# Here's the straightforward ansatz: multiple of rhs
# is not a solution of the homogeneous ODE, but creates
# a new linearly independent function: cos(t).
#----------------------------------------------------
> x_p:=A*t*sin(t);

> eval(subs(x(t)=x_p,ode_h));

> eval(subs(x(t)=x_p,ode));

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#----------------------------------------------------
# ... so we should add cos(t), but this won't work
# because it solves the homogeneous ODE, so multiply
# by t first and then add:
#----------------------------------------------------
> x_p:=A*t*sin(t)+B*t*cos(t);

> eval(subs(x(t)=x_p,ode_h));

> eval(subs(x(t)=x_p,ode));

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#----------------------------------------------------
# Of course, that now simply produces sin(t), which
# we ought to add to the ansatz. But it's a solution
# of the homogeneous ODE, so multiply by t and then
# add. But, hang on, that's what we started with.
# AAAARGH.
#
# Solution: Need to increase the power of t:
#----------------------------------------------------
> x_p:=A*t*sin(t)+B*t^2*cos(t);

> eval(subs(x(t)=x_p,ode_h));

> eval(subs(x(t)=x_p,ode));

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#----------------------------------------------------
# ...and this works for B=1/8 and A=-B
#----------------------------------------------------
> subs(A=-1/8,B=1/8,eval(subs(x(t)=x_p,ode)));

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> > #---------------------------------------------------- # The ODE (second order correction for weakly damped # mass spring damper system) #---------------------------------------------------- > ode:=diff(x(t),t$2)+x(t)=-1/2tsin(t);

> #---------------------------------------------------- # ...and its homogeneous counterpart #---------------------------------------------------- > ode_h:=diff(x(t),t$2)+x(t);

> #---------------------------------------------------- # Solve the bloody thing using maple #---------------------------------------------------- > dsolve(ode,x(t));

> > #---------------------------------------------------- # Now do it "by hand". #---------------------------------------------------- > #---------------------------------------------------- # Here's the straightforward ansatz: multiple of rhs # is not a solution of the homogeneous ODE, but creates # a new linearly independent function: cos(t). #---------------------------------------------------- > x_p:=Atsin(t);

> eval(subs(x(t)=x_p,ode_h));

> eval(subs(x(t)=x_p,ode));

> > > #---------------------------------------------------- # ... so we should add cos(t), but this won't work # because it solves the homogeneous ODE, so multiply # by t first and then add: #---------------------------------------------------- > x_p:=Atsin(t)+Btcos(t);

> eval(subs(x(t)=x_p,ode_h));

> eval(subs(x(t)=x_p,ode));

> > > #---------------------------------------------------- # Of course, that now simply produces sin(t), which # we ought to add to the ansatz. But it's a solution # of the homogeneous ODE, so multiply by t and then # add. But, hang on, that's what we started with. # AAAARGH. # # Solution: Need to increase the power of t: #---------------------------------------------------- > x_p:=Atsin(t)+Bt^2cos(t);

> eval(subs(x(t)=x_p,ode_h));

> eval(subs(x(t)=x_p,ode));

> > #---------------------------------------------------- # ...and this works for B=1/8 and A=-B #---------------------------------------------------- > subs(A=-1/8,B=1/8,eval(subs(x(t)=x_p,ode)));

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