Real & Distinct Roots — Question 3

Question 3 – Solution

Strategy. Separate eventual decay from the shape created by two modes with opposite coefficients.

Step 1: Solve for the two coefficients. The characteristic equation is $(r+2)(r+3)=0$, so $y=Ae^{-2t}+Be^{-3t}$. The data give $A+B=0$ and $-2A-3B=1$. Hence $$\boxed{y=e^{-2t}-e^{-3t}}.$$ Its value and derivative at zero are 0 and 1, and both exponential terms satisfy the equation.

Step 2: Prove positivity and decay. For $t>0$, $y=e^{-3t}(e^t-1)>0$. Both exponential terms tend to zero, so the positive response eventually decays to zero. Positivity does not require both mode coefficients to be positive.

Step 3: Locate and evaluate the only peak. The derivative factors as $$y'=e^{-3t}(3-2e^t).$$ It is positive before $t_m=\ln(3/2)$ and negative afterward. At that time $e^{-t_m}=2/3$, so $$\boxed{t_m=\ln(3/2)},\qquad \boxed{y(t_m)=4/9-8/27=4/27}.$$

Step 4: Check the total area independently. Direct integration gives $\int_0^\infty y\,dt=1/2-1/3=\boxed{1/6}$. Alternatively, integrate the equation on $[0,T]$: $$y'(T)-y'(0)+5(y(T)-y(0))+6\int_0^T y\,dt=0.$$ The explicit formula verifies $y(T),y'(T)\to 0$ and convergence of the integral. Passing to the limit gives $-1+6\int_0^\infty y\,dt=0$, confirming the result.

Negative roots control the eventual limit of each mode. Their combination can initially increase: this solution starts at zero with positive slope, reaches a peak, and only then decreases.