Can we convert first order ode to second order ode?

Question

Is there any possibility to convert first order differential equation to second order differential equation?

I have a system of first order differential equations as below and i need the right hand side of the second order form of this first order equation. (if possible)

$$\frac{du(t)}{dt}=[tu_{2}(t) ; 4u_{1}(t)^{3/2}]$$

Any help would be appreciated.

Thanks!

Answer 1

For simplicity, I don't want to write all those subscripts, so will re-write your system as:

$$\displaystyle x' = t y \\ y' = 4 t^{3/2}x$$

If we take the derivative of each equation and substitute, we have:

$$x'' = y + t y' = \dfrac{x'}{t} + t (4 t^{3/2} x) \\ y'' = 6 t^{1/2} x + 4 t^{3/2} x' = 6 t^{1/2}\left(\dfrac{y'}{4 t^{3/2}} \right)+ 4 t^{3/2} ( t y)$$

Of course, you can clean these up by simplifying.

Note that there are closed solutions to each of these, but they are quite nasty.