Real Eigenvalues — Question 3

Question 3 – Solution

Strategy. Growing modal amplitudes can combine with opposite signs; compute coordinate derivatives before making a monotonicity claim.

Step 1: Solve in an eigenbasis. Eigenpairs are $2,(1,0)^T$ and $1,(-1,1)^T$. The initial vector is their sum, so $$\boxed{x=e^{2t}-e^t,\qquad y=e^t.}$$ Both eigenvalues are positive, making the origin an unstable node (a source). Every solution tends to zero backward in time, not forward.

Step 2: Recover the geometric branch. Since $y=e^t>0$, the phase curve is $\boxed{x=y^2-y,\ y>0}$. It is traversed toward increasing $y$. The limiting origin is excluded; the finite-time point with $x=0$ is $(0,1)$. The curve’s leftmost point has a vertical tangent, not an equilibrium.

Step 3: Locate coordinate reversal. We have $x'=e^t(2e^t-1)$, zero only at $t=-\ln 2$. It is negative before this time and positive after it. Therefore the unique global minimum is $\boxed{x=-1/4}$, at $y=1/2$. For $t<0$, $x=e^t(e^t-1)<0$; it vanishes only at $t=0$ and is positive for $t>0$. The limit $x\to 0$ as $t\to-\infty$ is not another zero.

Step 4: Distinguish modal and coordinate behavior. Backward, $X/e^t\to(-1,1)^T$, so the approach line is $y=-x$. Forward, $X/e^{2t}\to(1,0)^T$, giving the positive horizontal direction. Positive eigenvalues describe growth of modes with fixed coefficients; they do not force each original coordinate to increase. Here the two contributions to $x$ have opposite signs, producing a genuine reversal. The faster eigenmode eventually dominates because its coefficient is nonzero.