# Abdullah's question via email about Runge Kutta scheme

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In summary, the conversation discusses the process of writing a differential equation as a system of first order DEs and using the Runge Kutta scheme to solve it. The system consists of two equations, one for $u$ and one for $v$, with initial conditions given. After two steps with a step size of 0.1, the value of $y$ at $x=1.2$ is approximated to be 3.28752.
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First we need to write the DE as a system of first order DEs.

Let $\displaystyle u = y$ and $\displaystyle v = y'$. Then

\displaystyle \begin{align*} x^2\,y'' - 5\,x\,v + 7\,u &= 2\,x^3\ln{\left( x \right) }\\ x^2\,y'' &= 5\,x\,v - 7\,u + 2\,x^3\ln{ \left( x \right) } \\ y'' &= \frac{5\,x\,v - 7\,u + 2\,x^3\ln{\left( x \right) } }{x^2} \end{align*}

So the system is

\displaystyle \begin{align*} u' &= v , \quad \quad \quad \quad \quad \quad\quad\quad \quad \quad \quad u\left( 1 \right) = 5 \\ v' &= \frac{5\,x\,v - 7\,u + 2\,x^3\ln{\left( x \right) } }{x^2} , \quad\, v\left( 1 \right) = 2 \end{align*}

I have used my CAS to apply the Runge Kutta scheme. Here $\displaystyle f\left( x,u,v \right) = v$ and $\displaystyle g\left( x, u, v \right) = \frac{5\,x\,v - 7\,u + 2\,x^3\ln{ \left( x \right) } }{x^2}$.

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After having gone through two steps of $\displaystyle h = 0.1$, we arrive at $\displaystyle x = 1.2$. Since we let $\displaystyle u = y$, that means $\displaystyle y\left( 1.2 \right) = u_2 = 3.28752$.

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Great job breaking down the differential equation into a system of first order DEs! This is a common approach in solving more complex DEs. Your use of the Runge Kutta scheme is also a good choice for solving this type of system. It's important to note that the Runge Kutta method is an approximation technique and may not give an exact solution, but it can provide a good estimate. Keep up the good work!

## 1. What is the Runge Kutta scheme?

The Runge Kutta scheme is a numerical method used to solve ordinary differential equations. It is a type of iterative algorithm that approximates the solution to the differential equation by breaking it down into smaller steps.

## 2. How does the Runge Kutta scheme work?

The Runge Kutta scheme works by using a weighted average of several estimates of the slope at different points within a given interval. This allows for a more accurate approximation of the solution to the differential equation compared to other numerical methods.

## 3. What are the advantages of using the Runge Kutta scheme?

The Runge Kutta scheme is a very versatile and accurate numerical method for solving differential equations. It is also relatively easy to implement and can handle a wide range of differential equations, including stiff equations that are difficult to solve using other methods.

## 4. Are there any limitations to the Runge Kutta scheme?

While the Runge Kutta scheme is a powerful numerical method, it does have some limitations. It can be computationally expensive for systems with a large number of equations, and it may not always provide the most accurate results for highly nonlinear or discontinuous systems.

## 5. How is the Runge Kutta scheme used in scientific research?

The Runge Kutta scheme is commonly used in various fields of science and engineering, such as physics, biology, and economics. It allows researchers to model and simulate complex systems that involve differential equations, providing valuable insights and predictions for real-world phenomena.

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