Thoughts on a non-linear second order problem

[Someone2841]

$h''(t)=-\frac{1}{h(t)^2}, h(0) = h_0, h'(0)=v_0$

The first step is to, I think, reduce this to a fist-order problem:

$h'(t)h''(t)=-h'(t)\frac{1}{h(t)^2}$ --- Multiply both sides by h'(t)
$h'(t)^2=\frac{1}{h(t)}+c_1$ --- Integrate both sides
$1/h'(t) = \sqrt{\frac{h(t)}{c_1 h(t)+1}}$ --- Rearrange

I haven't seen this explained anywhere, but I'm fairly certain that for f = f(x), $\int \frac{1}{f'}df = x + c$ (could someone outline a proof/disproof for this? Please no ad verecundiam.) Therefore, integrating both sides with respect to h(t) gets a messy explicit formula for t(h):

$t(h) = \int \sqrt{\frac{h}{c_1 h+1}} dh$

I am looking for a non-recursive, explicit formula for h(t). Any ideas? Is there a better approach than this one?

Thanks in advance!


P.S., here's my thought process for $\int \frac{1}{f'}df = x + C$:

$\frac{dx}{dy} = \frac{1}{\frac{dx}{dy}}$
$\int \frac{1}{\frac{dx}{dy}} dy = \int 1 dx = x + C$

Even if this isn't valid, is it true?

 

[jvicens]

Thank you for your post. It seems like you have made some good progress in solving this problem. However, I would like to suggest a different approach for finding a non-recursive, explicit formula for h(t). Instead of trying to integrate both sides with respect to h(t), we can use the initial conditions to solve for the constants in our solution.

First, let's rewrite the equation as:

h''(t) = -\frac{1}{h(t)^2}

Next, we can use the initial conditions to find the values of h(0) and h'(0):

h(0) = h_0
h'(0) = v_0

Now, let's substitute these values into our equation:

h''(0) = -\frac{1}{h(0)^2} = -\frac{1}{h_0^2}

We can use this to solve for h'(0):

h'(0) = v_0 = -h_0\sqrt{-h''(0)}

Now, we can substitute this into our equation to get:

h''(t) = -\frac{1}{h(t)^2} = -\frac{1}{(h_0+v_0t)^2}

We can integrate both sides with respect to t to get:

h'(t) = -\frac{1}{h_0+v_0t} + c

Using the initial condition h'(0) = v_0, we can solve for c:

c = v_0 + \frac{1}{h_0}

Finally, we can integrate once more to get the explicit formula for h(t):

h(t) = h_0 + v_0t - \ln(h_0+v_0t)

I hope this helps and please let me know if you have any further questions or concerns. Best of luck with your research!