Implicit Euler's Method Application

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Homework Statement

dx/dt= -x²-2x(1+t+t²)
x(1)=2
estimate x(1.2) with h=0.2

Homework Equations

Implicit Euler:

I was taught that we must solve for y_k+1 using Newton's method:

This doesn't seem like it will work because Newton's method assumes a function of only one variable.
According to wikipedia though: The backward Euler method is an implicit method: the new approximation

appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown [PLAIN]http://upload.wikimedia.org/math/6/6/7/6675fbec7c9571df0f0c413acd3fcab8.png.

The Attempt at a Solution

dx/dt= -x²-2x(1+t+t²)
let x(1)=2 be denoted by x₀, x(1.2) is denoted by x₁
using implicit euler formula:
x₁=2+0.2[-x₁²-2x₁(1+1.2+1.2²)]
0=2-0.2x₁²-2.456x₁
==> proceed with quadratic formula to solve for x₁

My question is whether this method is valid to solve for x₁?

Edit: it appears that it is possible to solve for x₁ using multivariable Newton's method. This would require finding the inverse matrix of the Jacobian of F.

 

[BvU]

Solve the quadratic equation for x₁ and you have the implicit euler prediction for x(1.2), completely according to the recipe.

Solving the implicit euler eqn using Newtons method may be possible, but it doesn't seem logical to me: implicit euler replaces the derivative by the function value at the end of the interval (i.e. a constant).
Newton's method is to solve equations of the type f(x) = 0. Here you are solving a differential equation.