Numerical Method20775 marksNumerical: Euler's method ODEEuler's method and its accuracy
Approximate the solution of y' = 2x + y, y(0) = 1 using Euler's method with step size 0.1. Approximate the value of y(0.4).
Approximate the solution of $y' = 2x + y$, $y(0) = 1$ using Euler's method with step size 0.1. Approximate the value of $y(0.4)$. [5]
- ODE: $y' = f(x,y) = 2x + y$ - Initial condition: $y(0) = 1$, so $x0 = 0$, $y0 = 1$ - Step size: $h = 0.1$ - Target: $y(0.4)$ (requires 4 steps) $$y{n+1} = yn + h \cdot f(xn, yn), \qquad f(xn,yn) = 2xn + yn$$ Step 1: $x0 = 0,\ y0 = 1$ $$f(0, 1) = 2(0) + 1 = 1$$ $$y1 = 1 + 0.1(1) = 1.1$$ Step 2: ...