I have the following differential equation:
$y'=xy^2-(y/x)$ with the initial value $y(1)=1$.
I'm trying to get a 4-digit approximation of this differential equation for $y(1.5)$ but I have thus far been unable to understand the RK4 method. I have been trying to understand how you would solve this without a computing system and so far I have been unable to figure this out.
Anything would help. Thanks!
You set $h=0.5$. Then compute per cook-book recipe using rounding to 2 digits \begin{align} k_1&=hf(x,y)&&=0.5f(1,1)&&=0\\ k_2&=hf(x+0.5h, y+0.5k_1)&&=0.5f(1.25, 1)&&=0.22\\ k_3&=hf(x+0.5h, y+0.5k_2)&&=0.5f(1.25, 1.1)&&=0.5(1.25\cdot 1.21-1.1⋅0.8)=0.32\\ k_2&=hf(x+h, y+k_3)&&=0.5f(1.5,1.32)&&=0.5(2.73-0.9)=0.87\\ \\ y(1.5)&\approx 1+(k_1+2k_2+2k_3+k_4)/6&&=1+1.95/6&&=1.32 \end{align}
The double computation gives
k1 = 0.0
k2 = 0.225
k3 = 0.32853515625
k4 = 0.8809091939608258
1+(k1+2k2+2k3+k4)/6 = 1.331329917743471
The exact solution results from $u(x)=\dfrac1{xy(x)}$, $u(1)=1$ $$ (xu)'=u+xu'=-\frac{y'}{y^2}=-x+u\\\implies u'=-1,\; u=2-x \\\implies y(x)=\frac{1}{2x-x^2} $$ so that $y(\frac32)=\frac43=1.3333...$
