Euler’s Method — Question 1

Question 1 – Solution

Strategy. Evaluate each slope at the current numerical point, then move both time and the approximation to the next node.

Step 1: Compute the four updates. The recurrence is $Y_{n+1}=Y_n+(t_n-Y_n)/4$. Its exact values are $$\begin{array}{c|c|c|c} n&t_n&Y_n&f(t_n,Y_n)\\\hline 0&0&1&-1\\ 1&1/4&3/4&-1/2\\ 2&1/2&5/8&-1/8\\ 3&3/4&19/32&5/32\\ 4&1&81/128&\text{not needed} \end{array}$$ Thus $\boxed{Y_4=81/128=0.6328125}$.

Step 2: Check against the exact IVP. Solving $y'+y=t$ with integrating factor $e^t$ gives $$\boxed{y=t-1+2e^{-t}},\qquad \boxed{Y_4-y(1)=\frac{81}{128}-\frac 2e\approx-0.102946}.$$ The exact formula has $y(0)=1$ and derivative $1-2e^{-t}=t-y$, so it is the correct comparison solution.

Step 3: Interpret the polygon. On $[t_n,t_{n+1}]$, the Euler segment has slope $f(t_n,Y_n)$. It is tangent to the solution through the numerical point $(t_n,Y_n)$, which generally differs from the original exact solution after the first step. The polygon is continuous but usually has corners at its nodes.

Step 4: Diagnose the time index. The correct first update is $1+(1/4)(0-1)=0.75$. The program instead uses $1+(1/4)(1/4-1)=0.8125$, mixing the new time with the old state. It is not the stated explicit Euler method, which requires both arguments from the same current node.