Laplace Transforms — Question 9

Question 9 – Solution

Strategy. Separate the polynomial encoding of initial data from the proper rational forcing term.

Step 1: Recover data uniquely. Write $P(s)=s^4+5s^2+4$. Polynomial division gives $$P(s)Y=s^3+7s-10+\frac{48}{s+3}.$$ For initial values $c_j=y^{(j)}(0)$, transformation of the equation gives $$PY=c_0s^3+c_1s^2+(c_2+5c_0)s+(c_3+5c_1)+\frac{A}{s+3}.$$ Thus $\boxed{A=48,\quad(c_0,c_1,c_2,c_3)=(1,0,2,-10)}$. Polynomial division has a unique polynomial part and remainder; the displayed triangular relation then determines every $c_j$ uniquely.

Step 2: Resolve the real modes. Real partial fractions yield $$Y=\frac{(2/5)s+22/15}{s^2+1} +\frac{(3/13)s-14/39}{s^2+4}+\frac{24/65}{s+3},$$ so $$\boxed{y=\frac 25\cos t+\frac{22}{15}\sin t +\frac 3{13}\cos(2t)-\frac 7{39}\sin(2t)+\frac{24}{65}e^{-3t}.}$$

Step 3: Check the recovered experiment. The oscillatory terms are annihilated by $P(D)$ and $P(-3)=130$, so the exponential contributes $130(24/65)e^{-3t}=48e^{-3t}$. At zero the four derivatives are $$\begin{aligned} y(0)&=2/5+3/13+24/65=1,\\ y'(0)&=22/15-14/39-72/65=0,\\ y''(0)&=-2/5-12/13+216/65=2,\\ y'''(0)&=-22/15+56/39-648/65=-10. \end{aligned}$$ Absolute convergence holds for $\operatorname{Re}s>0$.

Step 4: Prove the minimum homogeneous order. All five complex modes at $i,-i,2i,-2i,-3$ have nonzero coefficients. Their linear independence forces every annihilating polynomial to vanish at all five distinct roots. Hence the least monic operator is $\boxed{(D+3)(D^2+1)(D^2+4)}$, of order five. The fourth-order equation leaves the exponential as a nonzero forcing; annihilating that forcing adds the fifth factor.