Series Solutions — Question 9

Question 9 – Solution

Strategy. Recover constant coefficients from the earliest derivatives at which they enter, while keeping the scope of identification explicit.

Step 1: Recover the constants. The measured derivatives are $b_2=1$, $b_3=2$, $b_4=7$, $b_5=24$, where $b_n=y^{(n)}(0)$. The equation and its first two derivatives at zero give $$b_3=A,\qquad b_4=A^2+B,\qquad b_5=A^3+2AB+C.$$ Thus $\boxed{A=2,\ B=3,\ C=4}$. The triangular equations prove uniqueness.

Step 2: Check the information threshold. Through degree four, only $A,B$ are fixed; arbitrary $C$ gives the same earlier derivatives. Degree five first reveals $C$, so it is necessary and sufficient in general. The next derivative is $$b_6=2b_5+3b_4+4b_3=77,\qquad \boxed{a_6=77/720}.$$

Step 3: Extend the series and check minimal order. For ordinary coefficients, $$a_{n+3}=\frac{2(n+2)(n+1)a_{n+2}+3(n+1)a_{n+1}+4a_n} {(n+3)(n+2)(n+1)}\quad(n\geq 0).$$ The three seeds determine every coefficient. Equivalently, its constant companion system has the everywhere-convergent matrix exponential, so the series is entire. A homogeneous regular equation of order one or two with $y(0)=y'(0)=0$ would have only the zero solution by uniqueness, contradicting $y''(0)=1$. Thus the least possible monic homogeneous order is three.

Step 4: Show the model assumption matters. Let $y$ be the recovered solution and $y_\varepsilon=y+\varepsilon x^6$. For any nonzero $\varepsilon$, this is entire and has the same measured coefficients through degree five. Yet for $L=D^3-2D^2-3D-4$, $$Ly_\varepsilon=\varepsilon(120x^3-60x^4-18x^5-4x^6)\not\equiv 0.$$ A finite Taylor record identifies the equation within the stipulated model class; it does not establish that an arbitrary analytic function belongs to that class.