I want to analytically solve ∂ y ∂ t = − α y ,where initial conditions are y = 1 at t = 0and show the forward Euler method gets a smaller answer than the backward Euler method ∀ t &gt; 0, provided that 0 &lt; α 2 Δ t 2 &lt; 1Any help whatsoever will be appreciated!

Question

I want to analytically solve            ∂      y              ∂      t        =  −      α    y  ,where initial conditions are   y  =  1 at   t  =  0and show the forward Euler method gets a smaller answer than the backward Euler method   ∀   t  &amp;gt;  0, provided that 0   &amp;lt;      α    2     Δ      t    2     &amp;lt; 1Any help whatsoever will be appreciated!

neobuzdanio · Accepted Answer

At the core you have to show that for       |    x      |    &amp;lt;  1  1  +  x  ≤      e    x    ≤      1          1      −      x        .

I want to analytically solve del y/del t=−alpha y, where initial conditions are y=1 at t=0 and show the forward Euler method gets a smaller answer than the backward Euler method AA t>0, provided that 0 <alpha^2 triangle t^2 < 1

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