- #1
Odious Suspect
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This is from an example in Thomas's Classical Edition. The task is to find a solution to ##\frac{dy}{dx}=x+y## with the initial condition ##x=0; y=1##. He uses what he calls successive approximations.
$$y_1 = 1$$
$$\frac{dy_2}{dx}=y_1+x$$
$$\frac{dy_3}{dx}=y_2+x$$
...
$$\frac{dy_{n+1}}{dx}=y_n+x$$
I can easily follow the process, but I'm not seeing why I should consider each subsequent expression to provide a better approximation of ##y##. Is there an easy way to explain this?
$$y_1 = 1$$
$$\frac{dy_2}{dx}=y_1+x$$
$$\frac{dy_3}{dx}=y_2+x$$
...
$$\frac{dy_{n+1}}{dx}=y_n+x$$
I can easily follow the process, but I'm not seeing why I should consider each subsequent expression to provide a better approximation of ##y##. Is there an easy way to explain this?