Is Backward-Euler method considered the same as Runge Kutta $2^{\text{nd}}$ order method?

Question

I have a book that quotes:

Euler's method, Modified Euler's method and Runge's method are Runge-Kutta methods of first, second and third order respectively. The fourth-order Runge-Kutta method is method is most commonly used and is often referred to as 'Runge-Kutta method' or 'classical Runge-Kutta method'

Similary Wikipedia categorizes Backward-Euler's method as ' Implicit methods' under the list of Runge-Kutta methods and also mentions:

The backward Euler method is first order.

Now the problem is that the same book (from which I have taken the above quote) solves the below problem using a method that seems quite different(at least to me) from the Backward-Euler's method.

Consider the first order initial value problem $y'=y+2x-x^{2}$,$y(0)=1$,$(0\le x\le\infty)$ with exact solution $y(x)=x^2+e^x$. For $x=0.1$, what is solution obtained using a single iteration of the second-order Runge-Kutta method with step size $h=0.1$

The book then shows the solution using:

$$k_1=hf(x_0,y_0)$$ $$k_2=hf(x_0+h,y_0+k_1)$$ $$y_1=y_0+\frac{1}{2}(k_1+k_2)$$

Here $f$ denotes the differential equation i.e. $y'=f(x,y)=y+2x-x^{2}$. Using the above equations and initial value, it gets the result as $y_1=1.1145$.

I tried to calculate the vaule using Backward-Euler's method using:

$$y_{1}=y_{0}+hf(x_{1},y_{1})$$ and I get the result as $y_1=1.1322$, which is different from the solution given in the book.

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So I have the following questions:

1. Is Backward-Euler method considered the same as Runge-Kutta $2^{\text{nd}}$ order (RK2) method? If yes, is my book incorrect with the solution?
2. Is the method used in the book the actual Runge-Kutta $2^{\text{nd}}$ order method which is completely different from Backward-Euler's method?
3. In case my first question's answer is yes, how can a method be a Runge-Kutta $2^{\text{nd}}$ order (RK2) while also being a $1^{\text{st}}$ order in itself? (no need to answer if first question's answer is no)

I am really confused with the way the book used the name Backward Euler as RK2 but then used a different method to solve a question that wanted RK2. Please help me understand this.

Note: My book states Backward Euler as Modified Euler's method (In case it's not so obvious).

Answer 1

No, the first order implicit backwards Euler method is different from the second order explicit trapezoidal or Heun method.

In your question you name and describe both methods correctly.

The list in the first book quote only refers to explicit methods, at no point is there a reference to the implicit backward Euler method.

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To recap:

• Forwards Euler method: $$y_{k+1}=y_k+h\,f(x_k,y_k)$$
• Backwards Euler method: $$y_{k+1}=y_k+h\,f(x_{k+1},y_{k+1}),$$ which in general requires the solution of a non-linear equation.
• Implicit trapezoidal method: $$\frac{y_{k+1}-y_k}h=\frac{f(x_k,y_k)+f(x_{k+1},y_{k+1})}2,$$ which again requires the solution of an in general non-linear equation. It is noted for its time symmetry.
• Explicit trapezoidal method, modified Euler method, Heun's method: $$\frac{y_{k+1}-y_k}h=\frac{f(x_k,y_k)+f(x_{k+1},\tilde y_{k+1})}2,~~\text{ where }~~\tilde y_{k+1}=y_k+h\,f(x_k,y_k)$$ is a sufficiently accurate approximation of the implicit method. In the form of stages based on a Butcher tableau it is \begin{align}k_1&=h\,f(x_k,y_k),\\k_2&=h\,f(x_k+h,y_k+k_1),\\y_{k+1}&=y_k+\frac12(k_1+k_2).\end{align}