Numerical Method20785 marksNumerical: Heun's method ODEHeun's method
Solve the following differential equation for 1 leq x leq 2, taking h = 0.25 using Heun’s method. y'(x) + x2y = 3x, with y(1) = 1.
Solve the following differential equation for $1 \leq x \leq 2$, taking $h = 0.25$ using Heun’s method. $y'(x) + x^2y = 3x$, with $y(1) = 1$. [5]
- ODE: $y'(x) + x^2 y = 3x$, so $y'(x) = 3x - x^2 y = f(x,y)$ - Initial condition: $y(1) = 1$ - Step size: $h = 0.25$ - Interval: $1 \le x \le 2$ - Number of steps: $(2-1)/0.25 = 4$ $$y{i+1} = yi + \frac{h}{2}(m1 + m2)$$ where $m1 = f(xi, yi)$ and $m2 = f(xi + h,\ yi + h,m1)$. ---