Euler’s Method — Question 10

Question 10 – Solution

Strategy. Track the sign of the time increment and distinguish discretization error from amplification of uncertain terminal data.

Step 1: March backward explicitly. Writing $t_j=1-j/4$, Euler gives $$Y_{j+1}=Y_j-\tfrac 14(-2Y_j)=\tfrac 32Y_j.$$ After four steps, $$\boxed{Y(0)=\left(\frac 32\right)^4e^{-2} =\frac{81}{16}e^{-2}\approx 0.685135}.$$

Step 2: Compare with the exact benchmark. The exact solution is $y(t)=e^{-2t}$, giving $y(0)=1$. Thus the numerical reconstruction error is $$\boxed{Y(0)-y(0)=\frac{81}{16}e^{-2}-1\approx-0.314865}.$$

Each decreasing-time step amplifies the current value by $1.5$. The exact amplification over a quarter-unit backward step is $e^{1/2}>1.5$, explaining the numerical underestimate.

Step 3: Propagate a data perturbation. Linearity shows that replacing the terminal value by $e^{-2}+\delta$ changes the numerical reconstruction by $$\boxed{(81/16)\delta},$$ whereas exact backward evolution changes it by $\boxed{e^2\delta}$. With $N$ backward steps of size magnitude $1/N$, the numerical amplification is $(1+2/N)^N$, which tends to $e^2$, not to zero. Refinement improves the evolution calculation but does not eliminate sensitivity to data.

Step 4: Separate the uncertainty floor. The possible exact initial values form $$\boxed{[1-0.001e^2,\ 1+0.001e^2] \approx[0.992611,1.007389]}.$$ Its half-width is about $0.007389$, exceeding $0.005$. No single reconstruction from the reported value can be within $0.005$ of every possible true initial value in that interval. Reducing the step alone therefore cannot supply the requested guarantee; the terminal data would also need tighter uncertainty. Forward physical decay and backward data amplification describe different directions of evolution.