(Solved): 5. Derive Heun's method \[ x(t+h)=x(t)+\frac{1}{2}\left(F_{1}+F_{2}\right), \ ...

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5. Derive Heun's method \[ x(t+h)=x(t)+\frac{1}{2}\left(F_{1}+F_{2}\right), \] where \[ \left\{\begin{array}{l} F_{1}=h f(t, x) \\ F_{2}=h f\left(t+h, x+F_{1}\right) . \end{array}\right. \] as follows. (a) Advance one step using Taylor's method by including terms up to 2nd order. \[ x(t+h)=x(t)+h x^{\prime}(t)+\frac{h^{2}}{2 !} x^{\prime \prime}(t)+\frac{h^{3}}{3 !} x^{\prime \prime \prime}(t)+\frac{h^{4}}{4 !} x^{(4)}(t)+\cdots \] (b) Replace derivatives of \( x \) with those (partial derivatives) of \( f \). For this, assume \( x(t) \) solves the ODE \( x^{\prime}(t)=f(t, x(t)) \). (c) Replace partials of \( f \) with only evaluations of \( f \) using Taylor series of \( f(t+h, x+h f) \) in two variables. (d) Organize it.