How would you go about solving the following system of ODEs:
\begin{align*} & x''(t) - \frac{2}{y}x'(t) \ y'(t) = 0 \ & y''(t) + \frac{1}{y} \big(x'(t) - y'(t)\big) = 0 \end{align*}
Any help would be very much appreciated!
How would you go about solving the following system of ODEs:
\begin{align*} & x''(t) - \frac{2}{y}x'(t) \ y'(t) = 0 \ & y''(t) + \frac{1}{y} \big(x'(t) - y'(t)\big) = 0 \end{align*}
Any help would be very much appreciated!
From the first equation we can conclude :
\ln x'(t)= 2\ln y +C , so x'(t)=C_1\cdot y^2
plugging this into second equation gives :
y''(t)+\frac{1}{y}(C_1 \cdot y^2-y'(t))=0
Now substitute y'(t)=v , where $v$ is a function in terms of variable $y$ ,so:
y''(t)=v'_{y}\cdot v
Hence :
v'_{y}\cdot v+\frac{1}{y}(C_1 \cdot y^2-v)=0
this equation is equivalent to the first order non-linear ODE :
v'_y+C_1\cdot y \cdot v^{-1} -\frac{1}{y}=0
which can be solved using numerical methods .