# I have the following initial value problem on t in [0,T]: y′=y^2(1−elipson y) I wish to use the Trapezoidal method to solve this I.V.P., and I cannot use simply use the pblackictor-corrector method, i.e. improved Euler's method.

I have the following initial value problem on $t\in \left[0,T\right]$:
${y}^{\prime }={y}^{2}\left(1-ϵy\right)$
I wish to use the Trapezoidal method to solve this I.V.P., and I cannot use simply use the pblackictor-corrector method, i.e. improved Euler's method. The trapezoidal method is defined by:
${y}_{n+1}={y}_{n}+\frac{h}{2}\left({f}_{n+1}+{f}_{n}\right)$
Plugging in the value from the O.D.E., we have after some simplification:
${y}_{n+1}={y}_{n}+\frac{h}{2}\left({y}_{n+1}^{2}+{y}_{n}^{2}-ϵ{y}_{n+1}^{3}-ϵ{y}_{n}^{3}\right)$
This is an implicit method, so I must use a root finding method at each iteration to find the ${y}_{n+1}$ on the right side of the equation. I have read online that Newton's method is usually used for such purposes, but I cannot figure out how to implement it.
${y}_{n+1}={y}_{n}-\frac{f\left({y}_{n}\right)}{{f}^{\prime }\left({y}_{n}\right)}$
For example, what values would $f\left({y}_{n}\right)$ and ${f}^{\prime }\left({y}_{n}\right)$ take in this context? Thank you in advance!
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Bianca Chung
Your whole equation is $f$. More precisely, ${y}_{n+1}$ is the solution of
$0=f\left(y\right)=y-{y}_{n}-\frac{h}{2}\left({y}^{2}+{y}_{n}^{2}-ϵ\left({y}^{3}+{y}_{n}^{3}\right)\right)$
closest to ${y}_{n}$.

The derivative is then
${f}^{\prime }\left(y\right)=1-\frac{h}{2}\left(2y-3ϵ{y}^{2}\right)$
so that the Newton iteration is
${y}_{+}=y-\frac{f\left(y\right)}{{f}^{\prime }\left(y\right)}=\frac{{y}_{n}-\frac{h}{2}\left({y}^{2}-{y}_{n}^{2}-ϵ\left(2{y}^{3}-{y}_{n}^{3}\right)\right)}{1-\frac{h}{2}\left(2y-3ϵ{y}^{2}\right)}$