xấp xỉ nghiệm bằng phương pháp Euler

Walking a solution one small step at a time

• Not every differential equation can be solved with a formula — but you can still approximate the solution numerically.
• Euler's method 欧拉法 follows the slope field in tiny straight steps from a starting point.
• At each point, use the differential equation to get the slope, then step forward along it.
• Repeat, and you trace an approximate solution curve.

The update rule

• Given $\dfrac{dy}{dx}=f(x,y)$ and a start $(x_0,y_0)$, take a fixed step size 步长 $h$:
• $$y_{n+1}=y_n+h\,f(x_n,y_n),\qquad x_{n+1}=x_n+h$$
• The new $y$ = old $y$ + (slope there) × (step $h$).
• It's "current value plus slope times step" — a tangent-line jump.

Euler's method steps along the tangent; the estimate drifts from the true solution

Marching through a table

• Organize the work in a table: at each row, read $(x_n,y_n)$, compute the slope $f(x_n,y_n)$, then get $y_{n+1}$.
• Advance $x$ by $h$ each row and repeat until you reach the target $x$.
• Each step is just arithmetic; the differential equation supplies the slope.
• After a few rows you have an approximate $y$ at the target.

Accuracy and step size

• Euler's method is only an approximation — it follows straight tangents, so it drifts from the true curve.
• A smaller step size $h$ gives a more accurate estimate (but more steps).
• On a concave-up curve Euler tends to underestimate; concave-down, overestimate.
• More steps, less error — the classic accuracy trade-off.