Series Solutions — Question 8

Question 8 – Solution

Strategy. Let the failed coefficient equation identify a logarithmic term forced by resonance.

Step 1: Locate the formal obstruction. For $y=\sum a_nx^n$, matching $x^n$ gives $$(n+2)(n+1)(n-1)a_{n+2}=\begin{cases}1,&n=1,\\0,&n\ne 1.\end{cases}$$ At $n=0$ this requires $a_2=0$, but at $n=1$ it requires $0=1$. No formal ordinary power series, and hence no analytic solution, exists at zero.

Step 2: Solve the reduced equation. For $z=y''$, $(z/x)'=1/x$, giving $z=x\log x+Cx$. Integrating twice yields $$\boxed{y=A+Bx+\frac{x^3\log x}{6} +\left(\frac C6-\frac 5{36}\right)x^3.}$$ The three homogeneous modes are $1,x,x^3$; the forced logarithm is additional.

Step 3: Classify endpoint smoothness. As $x\downarrow 0$, $y\to A$, $y'\to B$ and $y''=x\log x+Cx\to 0$. These limits give a $C^2$ extension. But $y'''=\log x+1+C\to-\infty$, so no $C^3$ extension exists. Constants can change finite terms, not cancel the coefficient of $\log x$. This is a formal inconsistency, unlike a solvable but divergent coefficient recurrence.

Step 4: Verify the illustrated member. Set $A=B=0$, $C=5/6$. Then $y'=x^2(3\log x+1)/6$, so $$\boxed{x_{\min}=e^{-1/3},\qquad y_{\min}=-\frac 1{18e},\qquad y(1)=0,\quad\lim_{x\downarrow 0}y(x)=0.}$$ The derivative changes from negative to positive at this unique minimum. The open marker records the original domain $x>0$, even though a $C^2$ extension to zero is possible.