The 'Guessing Method' (Mathematical Model for a Simple Robot)

[Kemikeren]

Please see the attached details.


The questions is:



How do I solve this problem?

I have tried to solve the characteristic polynomial which gave me -1/2 with multiplicity 2. Thus giving me y(hom) = c1*exp(-1/2t)+c2*t*exp(-1/2t)

I now that the solution is y = y(hom)+y0 where I got y(0) = A*sin(2t)+B*cos(2t)

But I'm not quite sure where I went wrong.

Thank you in advance!

 

[HallsofIvy]

The "guessing method", referred to here, is probably what I would call "the method of undetermined coefficients" (that sounds better than "guessing"!). We have the second order, linear, non-homogeneous differential equation with constant coefficients, \(\displaystyle \frac{d^3x}{dt^3}+ (\alpha+ \beta)\frac{d^2x}{dt^2}+ (1+ \alpha\beta)\frac{dx}{dt}= sin(2t)\).

The "associated homogeneous equation" is \(\displaystyle \frac{d^3x}{dt^3}+ (\alpha+ \beta)\frac{d^2x}{dt^2}+ (1+ \alpha\beta)\frac{dx}{dt}= 0\). That has characteristic equation \(\displaystyle r^3+ (\alpha+ \beta)r^2+ (1+\alpha\beta)r= 0\). It's not too difficult to see that has roots 0, \(\displaystyle -\alpha\), and \(\displaystyle -\beta\) so the general solution to the associated homogeneous equation is \(\displaystyle x(t)= A+ Be^{-\alpha t}+ Ce^{-\beta t}\).

Now, since \(\displaystyle sin(2t)\) is of the form we would expect for a solution to a "linear differential equation with constant coefficients" (with characteristic value 2i), we "guess", or look for coefficients P and Q that will make Psin(2t)+
Qcos(2t) a solution to the entire equation. Calculate the first, second, and third derivatives of that, put them into the original equation and see what values of P and Q will make it true. That solution, added to the general solution to the associated homogeneous equation, gives the general solution to the entire equation.