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The dot product isBoth vectors have squared Euclidean norm $a^2+(a+1{)}^2=2a^2+2a+1$. Under the given condition $a^2+a=1/2$, their dot product is $1$ and each squared Euclidean norm is $2$. Hence

The chain rule givesFor this to vanish for every differentiable $f$, its coefficient must be zero:The resulting $f(x-t)$ is a profile transported to the right at unit speed by the transport equation.

Substitute $r=-x^2$ into the geometric series, without taking any derivatives:Thus the coefficient of $x^n$ in the Taylor series is $(-1{)}^{n/2}$ when $n$ is even and zero when $n$ is odd. If nonzero terms are numbered from one, the $n$th term is $(-1{)}^{n-1}{x}^{2n-2}$.