Undetermined Coefficients — Question 9

Question 9 – Solution

Strategy. Test closure under differentiation, then distinguish a method’s scope from the equation’s solvability.

Step 1: Select the admissible trial families. The root $-1$ has multiplicity three. For $f_1$, use $y_{p,1}=x^3e^{-x}(Ax^2+Bx+C)$. For $f_2=(1-\cos 2x)/2$, none of the frequencies $0,\pm 2i$ is a characteristic root, so use $y_{p,2}=a+b\cos 2x+c\sin 2x$. Both have finite derivative families. The third forcing does not, as shown below.

Step 2: Match and check the admissible cases. For $y=e^{-x}v$, the equation becomes $v'''=x^2$ in the first case. Thus $$\boxed{y_{p,1}=x^5e^{-x}/60.}$$ Its shifted third derivative is exactly $x^2$. In the second case, $$L(1)=1,\quad L(\cos 2x)=-11\cos 2x+2\sin 2x,\quad L(\sin 2x)=-2\cos 2x-11\sin 2x.$$ The coefficient equations are $a=1/2$, $-11b-2c=-1/2$, $2b-11c=0$. They give $$\boxed{y_{p,2}=\frac 12+\frac{11}{250}\cos 2x+\frac 1{125}\sin 2x.}$$ Each complete solution adds $e^{-x}(c_0+c_1x+c_2x^2)$.

Step 3: Prove infinite derivative dimension. Start with $P_0=1$. Differentiation gives $P_{k+1}=P_k'+2xP_k$. If $P_k$ has degree $k$ and leading coefficient $2^k$, the second term has degree $k+1$ and leading coefficient $2^{k+1}$, whereas the derivative has lower degree. Induction proves these claims for every $k$. Polynomials of distinct degrees are linearly independent; multiplying by the never-zero $e^{x^2}$ preserves independence. Hence no nonzero constant-coefficient operator annihilates $f_3$.

Step 4: Interpret the obstruction correctly. Every standard finite trial, even after multiplication by a resonance power, lies in a finite-dimensional differentiation-invariant family. Applying $L$ keeps it in that family. It therefore cannot produce $e^{x^2}$, whose derivative family is infinite-dimensional. This is a limitation of the specified method. The normalized equation has constant coefficients and smooth forcing on $\mathbb R$, so the linear IVP theorem still gives a unique global solution for any three initial data.