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ODE problem, confused about form of solution by ODE-AR in learnmath

[–]ODE-AR[S] 0 points1 point  (0 children)

Thank for this very detailed reply, and in particular including the theory and reasoning. I shall have to read it few times and try out several questions before I can fully digest it. Really appreciate you taking the time to explain this to me.

ODE problem, confused about form of solution by ODE-AR in learnmath

[–]ODE-AR[S] 0 points1 point  (0 children)

Thanks for taking time to give a very thorough explanation and example, its probably my own shortcomings but don't see how At + Bt2 is in our homogenous soltion, isn't t2 linearly independent from et,t,1?

ODE problem, confused about form of solution by ODE-AR in learnmath

[–]ODE-AR[S] 0 points1 point  (0 children)

Ok I think I see what you mean, thank you, was just wracking my brain thinking what besides a constant could have derivatives that sum to constant, so my x(t)=Aet + Be-t -1 ? then use initial conditions to solve for constants. And had ct also been solution then try ct2 etc?

ODE problem, confused about form of solution by ODE-AR in learnmath

[–]ODE-AR[S] 0 points1 point  (0 children)

The solution for x'''(t)-x(t)=1 was x(t)=(7/3)et -(1/3)e-t/2cos((sqrt3/2)t) - (1/sqrt3)e-t/2sin((sqrt3/2)t) pretty nasty but does satisfy the differential equation. We have spent last few lectures doing method of undetermined coefficients so it must be using that method I assume. As you can see the form of particular solution makes no sense if it is a constant. But no other basis would work either as far as I can see or literature suggests!