Fundamental Sets of Solutions — Question 7

Question 7 – Solution

Strategy. Specify the scalar field and impose real-valuedness on the entire function, not just its value at one point.

Step 1: Verify complex completeness. Each candidate has second derivative $-4$ times itself. Their data columns at zero are $(1,2i)$ and $(1,-2i)$, with determinant $-4i\ne 0$. The initial-data criterion applies over $\mathbb C$ as well: real and imaginary parts each satisfy the real equation and its uniqueness theorem.

Step 2: Impose reality. Since $\overline{z_+}=z_-$, the conjugate of $Az_++Bz_-$ is $\overline Bz_++\overline Az_-$. Independence implies equality to its conjugate exactly when $$\boxed{B=\overline A.}$$ This relation is sufficient too, since the two terms then are conjugates.

Step 3: Convert coordinates. Euler’s formula gives $$Az_++Bz_-=(A+B)\cos 2t+i(A-B)\sin 2t.$$ For real $a,b$, the inverse relations are $$\boxed{A=(a-ib)/2,\qquad B=(a+ib)/2.}$$ They automatically satisfy the reality condition. The exponentials are not themselves real-valued functions on $\mathbb R$, so they are not elements of the real-valued solution space. A real fundamental set is $\cos 2t,\sin 2t$.

Step 4: Match the data. The real representation has $a=1$ and $2b=-2$, hence $$\boxed{y=\cos 2t-\sin 2t =\tfrac 12(1+i)e^{2it}+\tfrac 12(1-i)e^{-2it}.}$$ Here $A+B=1$ and $2i(A-B)=-2$, verifying the value and slope. The conjugate coefficients ensure that the resulting solution is real everywhere.