Abdullah's question via email about Runge Kutta scheme

[Prove It]

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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 $.

 

[jvicens]

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!