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I have a differential equation $$ x'(t) = tx + t^4$$ with initial condition $ x(5)=3$. I am asked to find the estimates using the taylor series method from $o < t < 5$ with $h=0.01$ steps. I get that you have to use the formula $$x_1 = x_0 + hx_0' + \frac{h^2}{2!} x_0'' +\cdots $$but this is a recursion. If I was given initial condition $t$ at $0$, then I can start the recursion and keep going with this formula. If I was given the initial condition at $t=5$, how do I work backwards? Thanks

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  • $\begingroup$ Do you have to use Taylor, it's much easier when you use just the normal way! $\endgroup$ – Jan Dec 4 '15 at 12:43
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Forward

$t_n=t_0 + nh=nh$, $n=0,1,2,...$

$x_{n+1}=x(t_{n+1})=x(t_n+\delta t)=x(t_n+h)=x(t_n)+h x'(t_n)+O(h^2)\\=x(t_n)+h( t_nx(t_n)+t_n^4) + O(h^2)$

Backward

$t_n=t_0 - nh=5-nh$, $n=0,1,2,...$

$x_{n+1}=x(t_{n+1})=x(t_n-\delta t)=x(t_n-h)=x(t_n)-h x'(t_n)+O(h^2)\\=x(t_n)-h( t_nx(t_n)+t_n^4) + O(h^2)$

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